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Integral Calculus IIT JEE

Amit M Agarwal

Chapter 1

Indefinite Integral - all with Video Answers

Educators

Chapter Questions

08:15

Problem 1

Let $f(x)=\int \frac{x^{2}}{\left(1+x^{2}\right)\left(1+\sqrt{1+x^{2}}\right)} d x$ and $f(0)=0$. Then
$f(1)$ is equal to
(a) $\log _{e}(1+\sqrt{2})$
(b) $\log _{e}(1+\sqrt{2})-\frac{\pi}{4}$
(c) $\log _{e}(1+\sqrt{2})+\frac{\pi}{4}$
(d) None of these

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
03:29

Problem 2

If $\int f(x) d x=f(x)$, then $\int\{f(x)\}^{2} d x$ is equal to
(a) $\frac{1}{2}\{f(x)\}^{2}$
(b) $\{f(x)\}^{3}$
(c) $\frac{\{f(x)\}^{3}}{3}$
(d) $\{f(x)\}^{2}$

Pritesh Ranjan
Pritesh Ranjan
Numerade Educator
03:52

Problem 3

If $\int f(x) d x=F(x)$, then $\int x^{3} f\left(x^{2}\right) d x$ is equal to
(a) $\frac{1}{2}\left[x^{2}\{F(x)\}^{2}-\int\{F(x)\}^{2} d x\right]$
(b) $\frac{1}{2}\left[x^{2} F\left(x^{2}\right)-\int F\left(x^{2}\right) d\left(x^{2}\right)\right]$
(c) $\frac{1}{2}\left[x^{2} F(x)-\frac{1}{2} \int\{F(x)\}^{2} d x\right]$
(d) None of the above

Pritesh Ranjan
Pritesh Ranjan
Numerade Educator
02:10

Problem 4

If $n$ is an odd positive integer, then $\int\left|x^{n}\right| d x$ is
equal to
(a) $\left|\frac{x^{n+1}}{n+1}\right|+C$
(b) $\frac{x^{n+1}}{n+1}+C$
(c) $\frac{|x|^{n} x}{n+1}+C$
(d) None of these

Pritesh Ranjan
Pritesh Ranjan
Numerade Educator
05:54

Problem 5

Let $F(x)$ be the primitive of $\frac{3 x+2}{\sqrt{x-9}}$ w.r.t. $x$. If $F(10)=60$, then the value of $F(13)$ is
(a) 66
(b) 132
(c) 248
(d) 264

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
04:03

Problem 6

$\int\left(x^{x}\right)^{x}\left(2 x \log _{e} x+x\right) d x$ is equal to
(a) $x^{\left(x^{x}\right)}+C$
(b) $\left(x^{x}\right)^{x}+C$
(c) $x^{2} \cdot \log _{e} x+C$
(d) None of these

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
02:36

Problem 7

The value of $\int x \log x(\log x-1) d x$ is equal to
(a) $2(x \log x-x)^{2}+C$
(b) $\frac{1}{2}(x \log x-x)^{2}+C$
(c) $(x \log x)^{2}+C$
(d) $\frac{1}{2}(x \log x)^{3}+C$

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
06:02

Problem 8

$\int \frac{x^{2}-1}{x^{3} \sqrt{2 x^{4}-2 x^{2}+1}} d x$ is equal to
(a) $\frac{\sqrt{2 x^{4}-2 x^{5}+1}}{x^{2}}+C$
(b) $\frac{\sqrt{2 x^{4}-2 x^{2}+1}}{x^{3}}+C$
(c) $\frac{\sqrt{2 x^{4}+2 x^{2}+1}}{x}+C$
(d) $\frac{\sqrt{2 x^{4}-2 x^{2}+1}}{2 x^{2}}+C$

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
02:26

Problem 9

Let $f(x)$ be a polynomial satisfying $f(0)=2, f^{\prime}(0)=3$ and $f^{\prime \prime}(x)=f(x)$. Then $f(4)$ is equal to
(a) $\frac{5\left(e^{8}+1\right)}{2 e^{4}}$
(b) $\frac{5\left(e^{8}-1\right)}{2 e^{4}}$
(c) $\frac{2 e^{4}}{5\left(e^{8}-1\right)}$
(d) $\frac{2 e^{4}}{5\left(e^{8}+1\right)}$

Naman Kumar
Naman Kumar
Numerade Educator
07:00

Problem 10

$\int \frac{e^{\left(x^{2}+4 \ln x\right)}-x^{3} e^{x^{2}}}{x-1} d x$ is equal to
(a) $\left(\frac{e^{3 \ln x}-e^{\ln x}}{2 x}\right) e^{x^{2}}+C$
(b) $\frac{(x-1) x e^{x^{2}}}{2}+C$
(c) $\frac{\left(x^{2}-1\right)}{2 x}-e^{x^{2}}+C$
(d) None of these

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
04:12

Problem 11

$\int \tan ^{4} x d x=A \tan ^{3} x+B \tan x+f(x)$, then
(a) $A=\frac{1}{3}, B=-1, f(x)=x+C$
(b) $A=\frac{2}{3}, B=-1, f(x)=x+C$
(c) $A=\frac{1}{3}, B=1, f(x)=x+C$
(d) $A=\frac{2}{3}, B=1, f(x)=-x+C$

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
01:39

Problem 12

If the anti-derivative of $\int \frac{\sin ^{4} x}{x} d x$ is $f(x)$, then $\int \frac{\sin ^{4}\{(p+q) x\}}{x} d x$ in terms of $f(x)$ is
(a) $f\{(p+q) x\}$
(b) $\frac{f\{(p+q) x\}}{p+q}$
(c) $f\{(p+q) x\}(p+q)$
(d) None of these

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
04:17

Problem 13

$\int\left(\frac{\sin \theta}{\cos 3 \theta}+\frac{\sin 3 \theta}{\cos 9 \theta}+\frac{\sin 9 \theta}{\cos 27 \theta}\right) d \theta$ is equal to
(a) $\frac{1}{2} \log \left|\frac{\sec 27 \theta}{\sec \theta}\right|+C$
(b) $\frac{1}{2} \log \left|\frac{\sec \theta}{\sec 27 \theta}\right|+C$
(c) $\frac{1}{2} \log \left|\frac{\sqrt[22]{\sec 27 \theta}}{\sec \theta}\right|+C$
(d) None of these

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
05:16

Problem 14

Let $x^{2} \neq n \pi-1, n \in N$. Then, the value of $\int x \sqrt{\frac{2 \sin \left(x^{2}+1\right)-\sin 2\left(x^{2}+1\right)}{2 \sin \left(x^{2}+1\right)+\sin 2\left(x^{2}+1\right)}} d x$ is equal to
(a) $\log \left|\frac{1}{2} \sec \left(x^{2}+1\right)\right|+C$
(b) $\log \left|\sec \left(\frac{x^{2}+1}{2}\right)\right|+C$
(c) $\frac{1}{2} \log \left|\sec \left(x^{2}+1\right)\right|+C$
(d) None of these

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
01:58

Problem 15

$\int \frac{d x}{\cos (2 x) \cos (4 x)}$ is equal to
(a) $\frac{1}{2 \sqrt{2}} \log \left|\frac{1+\sqrt{2} \sin 2 x}{1-\sqrt{2} \sin 2 x}\right|-\frac{1}{2}(\log |\sec 2 x-\tan 2 x|)+C$
(b) $\frac{1}{2 \sqrt{2}} \log \left|\frac{1+\sqrt{2} \sin 2 x}{1+\sqrt{2} \sin x}\right|-\frac{1}{2}(\log |\sec 2 x-\tan 2 x|)+C$
(c) $\frac{1}{\sqrt{2}} \log \left|\frac{1+\sqrt{2} \sin 2 x}{1-\sqrt{2} \sin 2 x}\right|-\frac{1}{2}(\log |\sec 2 x-\tan 2 x|)+C$
(d) None of the above

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:21

Problem 16

$\int \frac{1-7 \cos ^{2} x}{\sin ^{7} x \cos ^{2} x} d x=\frac{f(x)}{(\sin x)^{7}}+C$, then $f(x)$ is equal to
(a) $\sin x$
(b) $\cos x$
(c) $\tan x$
(d) $\cot x$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
06:48

Problem 17

$\int \frac{\sin ^{3} x}{\left(\cos ^{4} x+3 \cos ^{2} x+1\right) \tan ^{-1}(\sec x+\cos x)} d x$ is equal to
(a) $\tan ^{-1}(\sec x+\cos x)+C$
(b) $\log _{e}\left|\tan ^{-1}(\sec x+\cos x)\right|+C$
(c) $\frac{1}{(\sec x+\cos x)^{2}}+C$
(d) None of these

Sandip Ranjan
Sandip Ranjan
Numerade Educator
02:51

Problem 18

The primitive of the function $f(x)=x|\cos x|$, when $\frac{\pi}{2}<x<\pi$ is given by
(a) $\cos x+x \sin x+C$
(b) $-\cos x-x \sin x+C$
(c) $x \sin x-\cos x+C$
(d) None of the above

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
03:32

Problem 19

The primitive of the function $f(x)=(2 x+1)|\sin x|$, when $\pi<x<2 \pi$ is
(a) $-(2 x+1) \cos x+2 \sin x+C$
(b) $(2 x+1) \cos x-2 \sin x+C$
(c) $\left(x^{2}+x\right) \cos x+C$
(d) None of the above

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
03:30

Problem 20

Given, $f(x)=\left|\begin{array}{ccc}0 & x^{2}-\sin x & \cos x-2 \\ \sin x-x^{2} & 0 & 1-2 x \\ 2-\cos x & 2 x-1 & 0\end{array}\right|$, then
$\int f(x) d x$ is equal to
(a) $\frac{x^{3}}{3}-x^{2} \sin x+\sin 2 x+C$
(b) $\frac{x^{3}}{3}-x^{2} \sin x-\cos 2 x+C$
(c) $\frac{x^{3}}{3}-x^{2} \cos x-\cos 2 x+C$
(d) None of the above

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
04:13

Problem 21

$\int \frac{d x}{(x+1)(x-2)}=A \log (x+1)+B \log (x-2)+C$, where
(a) $A+B=0$
(b) $A B=0$
(c) $A / B=-1$
(d) None of these

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
04:51

Problem 22

If $\int \frac{d x}{\left(x^{2}+1\right)\left(x^{2}+4\right)}=k \tan ^{-1} x+l \tan ^{-1} \frac{x}{2}+C$, then
(a) $k=\frac{1}{3}$
(b) $l=\frac{2}{3}$
(c) $k=-\frac{1}{3}$
(d) $l=-\frac{1}{6}$

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
07:28

Problem 23

If $\int x \log \left(1+x^{2}\right) d x=\phi(x) \log \left(1+x^{2}\right)+x(\psi)+C$, then
(a) $\phi(x)=\frac{1+x^{2}}{2}$
(b) $\psi(x)=\frac{1+x^{2}}{2}$
(c) $\psi(x)=-\frac{1+x^{2}}{2}$
(d) $\phi(x)=-\frac{1+x^{2}}{2}$

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
10:57

Problem 24

If $\int \frac{4 e^{x}+6 e^{-x}}{9 e^{x}-4 e^{-x}} d x=A x+B \log _{e}\left(9 e^{2 x}-4\right)+C$, then
(a) $A=\frac{3}{2}$
(b) $B=\frac{35}{36}$
(c) $C$ is indefinite
(d) $A+B=-\frac{19}{36}$

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
05:13

Problem 25

If $\int \tan ^{5} x d x=A \tan ^{4} x+B \tan ^{2} x+g(x)+C$, then
(a) $A=\frac{1}{4}, B=-\frac{1}{2}$
(b) $g(x)=\ln |\sec x|$
(c) $g(x)=\ln |\cos x|$
(d) $A=-\frac{1}{4}, B=\frac{1}{3}$

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
02:18

Problem 26

Statement I If $y$ is a function of $x$ such that $y(x-y)^{2}=x$, then $\int \frac{d x}{x-3 y}=\frac{1}{2}\left[\log (x-y)^{2}-1\right]$
Statement II $\int \frac{d x}{x-3 y}=\log (x-3 y)+C$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:55

Problem 27

Statement I Integral of an even function is not always an odd function.
Statement II Integral of an odd function is an even function.

Linh Vu
Linh Vu
Numerade Educator
02:43

Problem 28

Statement I If $a>0$ and $b^{2}-4 a c<0$, then the value of the integral $\int \frac{d x}{a x^{2}+b x+c}$ will be of the type $\mu \tan ^{-1} \frac{x+A}{B}+C$, where $A, B, C, \mu$ are constants.
Statement II If $a>0, b^{2}-4 a c<0$, then $a x^{2}+b x+C$ can be written as sum of two squares.

Gaurav Kalra
Gaurav Kalra
Numerade Educator
02:48

Problem 29

$$
\begin{aligned}
&\text { Statement I } \int\left(\frac{1}{1+x^{4}}\right) d x=\tan ^{-1}\left(x^{2}\right)+C \\
&\text { Statement II } \int \frac{1}{1+x^{2}} d x=\tan ^{-1} x+C
\end{aligned}
$$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
03:23

Problem 30

$$
\begin{aligned}
&\text { Statement I } \int 2^{\tan ^{-1} x} d\left(\cot ^{-1} x\right)=\frac{2^{\tan ^{-1} x}}{\ln 2}+C \\
&\text { Statement II } \frac{d}{d x}\left(a^{x}+C\right)=a^{x} \ln a
\end{aligned}
$$

Naman Kumar
Naman Kumar
Numerade Educator
02:20

Problem 31

If $I=\int \frac{d x}{x-\sqrt{9 x^{2}+4 x+6}}$ to evaluate $I$, one of the most proper substitution could be
(a) $\sqrt{9 x^{2}+4 x+6}=u \pm 3 x$
(b) $\sqrt{9 x^{2}+4 x+6}=3 u \pm x$
(c) $x=\frac{1}{t}$
(d) $9 x^{2}+4 x+6=\frac{1}{t}$

Adrian Co
Adrian Co
Numerade Educator
06:02

Problem 32

$\int \frac{\left(x+\sqrt{1+x^{2}}\right)^{15}}{\sqrt{1+x^{2}}} d x$ is equal to
(a) $\frac{\left(x+\sqrt{1+x^{2}}\right)^{16}}{10}+C$
(b) $\frac{1}{15\left(\sqrt{1+x^{2}+x}\right)}+C$
(c) $\frac{15}{\left(\sqrt{1+x^{2}}-x\right)}+C$
(d) $\frac{\left(x+\sqrt{1+x^{2}}\right)^{15}}{15}+C$

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
06:02

Problem 33

To evaluate $\int \frac{d x}{(x-1) \sqrt{-x^{2}+3 x-2}}$ one of the most suitable substitution could be
(a) $\sqrt{-x^{2}+3 x-2}=u$
(b) $\sqrt{-x^{2}+3 x-2}=(u x \sqrt{2})$
(c) $\sqrt{-x^{2}+3 x-2}=u(1-x)$
(d) $\sqrt{-x^{2}+3 x-2}=u(x+2)$

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
05:33

Problem 34

The relation between $I_{4,2}$ and $I_{2,2}$ is
(a) $I_{4,2}=\frac{1}{6}\left(-\sin ^{3} x \cos ^{3} x+3 I_{2,2}\right)$
(b) $I_{4,2}=\frac{1}{6}\left(\sin ^{3} x \cos ^{3} x+3 I_{2,2}\right)$
(c) $I_{4,2}=\frac{1}{6}\left(\sin ^{3} x \cos ^{3} x-3 I_{2,2}\right)$
(d) $I_{4,2}=\frac{1}{4}\left(-\sin ^{3} x \cos ^{3} x+2 I_{2,2}\right)$

Anas Venkitta
Anas Venkitta
Numerade Educator
04:11

Problem 35

The relation between $I_{4,2}$ and $I_{6,2}$ is
(a) $I_{4,2}=\frac{1}{5}\left(\sin ^{5} x \cos ^{3} x+8 I_{6,2}\right)$
(b) $I_{4,2}=\frac{1}{5}\left(-\sin ^{5} x \cos ^{3} x+8 I_{6,2}\right)$
(c) $I_{4,2}=\frac{1}{5}\left(\sin ^{5} x \cos ^{3} x-8 I_{6,2}\right)$
(d) $I_{4,2}=\frac{1}{6}\left(\sin ^{5} x \cos ^{3} x+8 I_{6,2}\right)$

Ankit Singh
Ankit Singh
Numerade Educator
01:43

Problem 36

$$
\begin{aligned}
&\text { The relation between } I_{4,2} \text { and } I_{4,4} \text { is }\\
&\text { (a) } I_{4,2}=\frac{1}{3}\left(\sin ^{5} x \cos ^{3} x+8 I_{4,4}\right)
\end{aligned}
$$
(b) $I_{4,2}=\frac{1}{3}\left(-\sin ^{5} x \cos ^{3} x+8 I_{4,4}\right)$
(c) $I_{4,2}=\frac{1}{3}\left(\sin ^{5} x \cos ^{3} x-8 I_{4,4}\right)$
(d) $I_{4,2}=\frac{1}{3}\left(\sin ^{5} x \cos ^{3} x+6 I_{4,4}\right)$

Dheeraj Sharma
Dheeraj Sharma
Numerade Educator
02:18

Problem 37

Which of the following is true for $f(x)$
(a) $f(x)$ is one-one and into
(b) $\{f(x)\}$ is non-periodic, where \{\} denotes fractional part of $x$.
(c) $f(x)=4$ has only two solutions.
(d) $f(x)=f^{-1}(x)$ has only one solution.

Aman Gupta
Aman Gupta
Numerade Educator
02:21

Problem 38

let $g(x)=\log _{e}(\sin x)$, and $\int f(g(x)) \cos x d x=h(x)+c$
(where $c$ is constant of integration), then $h\left(\frac{\pi}{2}\right)$ is equal
$\begin{array}{ll}\text { to } & \text { (b) } \frac{1}{2010}\end{array}$
(c) 1
(d) $\frac{1}{2011}$

Naman Kumar
Naman Kumar
Numerade Educator
03:16

Problem 39

The equation $f(x)=0$ has
(a) all four distinct roots
(b) three distinct real roots
(c) two real and two imaginary
(d) all four imaginary roots

Pritesh Ranjan
Pritesh Ranjan
Numerade Educator
02:27

Problem 40

The minimum value of $f(x)$ is
(a) $-136$
(b) $-100$
(c) $-84$
(d) $-68$

Pritesh Ranjan
Pritesh Ranjan
Numerade Educator
01:17

Problem 41

$\int \frac{g(x)}{f(x)} d x$, equals
(a) $\tan ^{-1}\left(\frac{x-2}{2}\right)+c$
(b) $\tan ^{-1}\left(\frac{x-1}{1}\right)+c$
(c) $\tan ^{-1}(x)+c$
(d) None of these

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
01:23

Problem 42

Match the following :

Hitendra Singh
Hitendra Singh
Numerade Educator
01:23

Problem 43

Match the following :

Hitendra Singh
Hitendra Singh
Numerade Educator
02:46

Problem 44

If $\int \frac{(2 x+3) d x}{x(x+1)(x+2)(x+3)+1}=C-\frac{1}{f(x)}$, where $f(x)$ is of the form of $a x^{2}+b x+c$, then $(a+b+c)$ equals to

Pritesh Ranjan
Pritesh Ranjan
Numerade Educator
05:54

Problem 45

Let $F(x)$ be the primitive of $\frac{3 x+2}{\sqrt{x-9}}$ w.r.t. $x$. If $F(10)=60$, then the sum of digits of the value of $F(13)$, is

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
01:43

Problem 46

Let $u(x)$ and $v(x)$ are differentiable function such that $\frac{u(x)}{v(x)}=7$. If $\frac{u^{\prime}(x)}{v^{\prime}(x)}=p$ and $\left(\frac{u(x)}{v(x)}\right)^{\prime}=q$, then $\frac{p+q}{p-q}$ has the value equal to ..........

Lucas Finney
Lucas Finney
Numerade Educator
00:33

Problem 46

$$
\text { If } \int \frac{1}{\left(x^{2}-1\right)} \ln \left(\frac{x-1}{x+1}\right) d x=6 A\left[\ln \left(\frac{x-1}{x+1}\right)\right]^{2}+C, \text { then find } 24 A \text { . }
$$

Amrita Bhasin
Amrita Bhasin
Numerade Educator
00:33

Problem 47

$$
\text { If } \int \frac{1}{\left(x^{2}-1\right)} \ln \left(\frac{x-1}{x+1}\right) d x=6 A\left[\ln \left(\frac{x-1}{x+1}\right)\right]^{2}+C, \text { then find } 24 A \text { . }
$$

Amrita Bhasin
Amrita Bhasin
Numerade Educator
02:38

Problem 48

$$
\text { If } \int \frac{e^{x}\left(2-x^{2}\right)}{(1-x) \sqrt{1-x^{2}}} d x=\mu e^{x}\left(\frac{1+x}{1-x}\right)^{\lambda}+C, \text { then } \chi(\lambda+\mu) \text { is equal to }
$$

Naman Kumar
Naman Kumar
Numerade Educator
04:27

Problem 49

$$
\begin{aligned}
&\text { If } \int \frac{\cos x-\sin x+1-x}{e^{x}+\sin x+x} d x=\ln \{f(x)\}+g(x)+C, \text { where }\\
&C \text { is the constant of integrating and } f(x) \text { is positive, then }\\
&\frac{f(x)+g(x)}{e^{x}+\sin x} \text { is equal to .......... }
\end{aligned}
$$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
02:23

Problem 50

$$
\begin{aligned}
&\text { Suppose } A=\int \frac{d x}{x^{2}+6 x+25} \text { and } B=\int \frac{d x}{x^{2}-6 x-27} \text { . }\\
&\text { If } 12(A+B)=\lambda \cdot \tan ^{-1}\left(\frac{x+3}{4}\right)+\mu \cdot \ln \left|\frac{x-9}{x+3}\right|+C, \text { then }\\
&\text { the value of }(\lambda+\mu) \text { is } \ldots
\end{aligned}
$$

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator
03:03

Problem 51

$$
\begin{aligned}
&\text { If } \int \frac{\cos 6 x+\cos 9 x}{1-2 \cos 5 x} d x=-\frac{\sin 4 x}{k}-\sin x+C, \text { then the }\\
&\text { value of } k \text { is } \ldots \ldots \ldots
\end{aligned}
$$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:55

Problem 52

$$
\begin{aligned}
&\text { The value of } \int \frac{\tan x}{1+\tan x+\tan ^{2} x} d x=x-\frac{1}{\sqrt{A}} \tan ^{-1}\\
&\left(\frac{2 \tan x+1}{\sqrt{A}}\right)+C, \text { then the value of } A \text { is.......... }
\end{aligned}
$$

Naman Kumar
Naman Kumar
Numerade Educator
00:52

Problem 53

$$
\begin{aligned}
&\int \sin ^{5 / 2} x \cos ^{3} x d x=2 \sin ^{A / 2} x\left[\frac{1}{B}-\frac{1}{C} \sin ^{2} x\right]+C\\
&\text { then the value of }(A+B)-C \text { is equal to } \ldots \ldots \ldots .
\end{aligned}
$$

Amrita Bhasin
Amrita Bhasin
Numerade Educator
03:40

Problem 54

$$
\begin{aligned}
&\text { If } \int\left(x^{2010}+x^{804}+x^{402}\right)\left(2 x^{1608}+5 x^{402}+10\right)^{1 / 402} d x \\
&=\frac{1}{10 a}\left(2 x^{2010}+5 x^{804}+10 x^{402}\right)^{a / 402} . \text { Then }(a-400) \text { is } \\
&\text { equal to .......... }
\end{aligned}
$$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
05:10

Problem 55

$$
\begin{aligned}
&\text { If } \int e^{x^{3}+x^{2}-1}\left(3 x^{4}+2 x^{3}+2 x\right) d x=h(x)+c \text { Then the }\\
&\text { value of } h(1) \cdot h(-1) \text { , is ......... }
\end{aligned}
$$

Harmender Singh Yadav
Harmender Singh Yadav
Numerade Educator
01:14

Problem 56

$$
\text { Evaluate } e^{(x \sin x+\cos x)} d x\left[\frac{x^{4} \cos ^{3} x-x \sin x+\cos x}{x^{2} \cos ^{2} x}\right]
$$

Wendi Zhao
Wendi Zhao
Numerade Educator
07:39

Problem 57

$$
\text { Evaluate } \int \sqrt{x+\sqrt{x^{2}+2}} d x
$$

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
01:14

Problem 58

$$
\text { Evaluate } \int \frac{d x}{\left(\sqrt{(x-\alpha)^{2}-\beta^{2}}\right)(a x+b)} .
$$

Amrita Bhasin
Amrita Bhasin
Numerade Educator
09:01

Problem 59

$$
\text { Evaluate } \int \frac{\sqrt{1+\sqrt[3]{x}}}{\sqrt[3]{x^{2}}} d x
$$

Vipender Yadav
Vipender Yadav
Numerade Educator
01:09

Problem 60

$$
\text { Evaluate } \int \frac{\sin ^{3}(\theta / 2) d \theta}{\cos \theta / 2 \sqrt{\cos ^{3} \theta+\cos ^{2} \theta+\cos \theta}}
$$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
04:00

Problem 61

$$
\text { Evaluate } \int \frac{(2 \sin \theta+\sin 2 \theta) d \theta}{(\cos \theta-1) \sqrt{\cos \theta+\cos ^{2} \theta+\cos ^{3} \theta}} .
$$

Patrick Vaughn
Patrick Vaughn
Numerade Educator
01:39

Problem 62

$$
\begin{aligned}
&\text { Connect } \int x^{m-1}\left(a+b x^{n}\right)^{p} d x \text { with } \\
&\int x^{m-n-1}\left(a+b x^{n}\right)^{p} d x \text { and evaluate } \int \frac{x^{8} d x}{\left(1-x^{3}\right)^{1 / 3}}
\end{aligned}
$$

Gregory Higby
Gregory Higby
Numerade Educator
01:34

Problem 63

$$
\text { Evaluate } \int \operatorname{cosec}^{2} x \ln (\cos x+\sqrt{\cos 2 x}) d x
$$

Steven Clarke
Steven Clarke
Numerade Educator
01:55

Problem 64

$$
\text { Evaluate } \int \frac{d x}{(\sin x+a \sec x)^{2}}, a \in N
$$

Linda Hand
Linda Hand
Numerade Educator
02:21

Problem 65

$$
\text { Evaluate } \int \frac{d x}{x-\sqrt{x^{2}+2 x+4}}
$$

Gregory Higby
Gregory Higby
Numerade Educator
13:10

Problem 66

$$
\text { Evaluate } \int \frac{d x}{1+\sqrt{x^{2}+2 x+2}} \cdot
$$

Sirat Shah
Sirat Shah
Numerade Educator
01:26

Problem 67

$$
\text { Evaluate } \int \frac{x^{4}+1}{x^{6}+1} d x
$$

Audrey Fong
Audrey Fong
Numerade Educator
03:34

Problem 68

$$
\text { Evaluate } \int \frac{d x}{\left(1-x^{3}\right)^{1 / 3}} d x
$$

Harmender Singh Yadav
Harmender Singh Yadav
Numerade Educator
01:49

Problem 69

$$
\text { Evaluate } \int \frac{\left(x+\sqrt{1+x^{2}}\right)^{15}}{\sqrt{1+x^{2}}} d x
$$

Pritesh Ranjan
Pritesh Ranjan
Numerade Educator
06:31

Problem 70

$$
\begin{aligned}
&\text { If } y^{2}=a x^{2}+2 b x+c, \text { and } u_{n}=\int \frac{x^{n}}{y} d x, \text { prove that }\\
&(n+1) a u_{n+1}+(2 n+1) b u_{n}+{ }^{n} c y_{n-1}=x^{n} y \text { and deduce }\\
&\text { that } a u_{1}=y-b u_{0} ; 2 a^{2} u_{2}=y(a x-3 b)-\left(a c-3 b^{2}\right) u_{0} \text { . }
\end{aligned}
$$

Willis James
Willis James
Numerade Educator
01:52

Problem 71

$\int \frac{\sec ^{2} x}{(\sec x+\tan x)^{9 / 2}} d x$ equals to (for some arbitrary
constant $K$ ) [Only One Correct Option 2012]
(a) $\frac{-1}{(\sec x+\tan x)^{1 / 2}}\left\{\frac{1}{11}-\frac{1}{7}(\sec x+\tan x)^{2}\right\}+K$
(b) $\frac{1}{(\sec x+\tan x)^{1 / 2}}\left\{\frac{1}{11}-\frac{1}{7}(\sec x+\tan x)^{2}\right\}+K$
(c) $\frac{-1}{(\sec x+\tan x)^{1 / 2}}\left\{\frac{1}{11}+\frac{1}{7}(\sec x+\tan x)^{2}\right\}+K$
(d) $\frac{1}{(\sec x+\tan x)^{1 / 2}}\left\{\frac{1}{11}+\frac{1}{7}(\sec x+\tan x)^{2}\right\}+K$

Nolwazi Dube
Nolwazi Dube
Numerade Educator
08:37

Problem 72

If $I=\int \frac{e^{x}}{e^{4 x}+e^{2 x}+1} d x, \exists=\int \frac{e^{-x}}{e^{-4 x}+e^{-2 x}+1} d x .$ Then,
for an arbitrary constant $c$, the value of $\exists-I$ equals [Only One Correct Option 2008]
(a) $\frac{1}{2} \log \left|\frac{e^{4 x}-e^{2 x}+1}{e^{4 x}+e^{2 x}+1}\right|+C$
(b) $\frac{1}{2} \log \left|\frac{e^{2 x}+e^{x}+1}{e^{2 x}-e^{x}+1}\right|+C$
(c) $\frac{1}{2} \log \left|\frac{e^{2 x}-e^{x}+1}{e^{2 x}+e^{x}+1}\right|+C$
(d) $\frac{1}{2} \log \left|\frac{e^{4 x}+e^{2 x}+1}{e^{4 x}-e^{2 x}+1}\right|+C$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
07:06

Problem 73

The integral $\int \frac{2 x^{12}+5 x^{9}}{\left(x^{5}+x^{3}+1\right)^{3}} d x$ is equal to
(a) $\frac{-x^{5}}{\left(x^{5}+x^{3}+1\right)^{2}}+C$
(b) $\frac{x^{10}}{2\left(x^{5}+x^{3}+1\right)^{2}}+C$
(c) $\frac{x^{5}}{2\left(x^{5}+x^{3}+1\right)^{2}}+C$
(d) $\frac{-x^{10}}{2\left(x^{5}+x^{3}+1\right)^{2}}+C$
where, $C$ is an arbitrary constant.

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
03:37

Problem 74

(a) $\left(\frac{x^{4}+1}{x^{4}}\right)^{\frac{1}{4}}+C$
(b) $\left(x^{4}+1\right)^{\frac{1}{4}}+C$
(c) $-\left(x^{4}+1\right)^{\frac{1}{4}}+C$
(d) $-\left(\frac{x^{4}+1}{x^{4}}\right)^{\frac{1}{4}}+C$

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator
01:33

Problem 75

The integral $\int\left(1+x-\frac{1}{x}\right) e^{x+\frac{1}{x}} d x$ is equal to
(a) $(x-1) e^{x+\frac{1}{x}}+C \quad$ (b)
$x e^{x+\frac{1}{x}}+C$
(c) $(x+1) e^{x+\frac{1}{x}}+C$
(d) $-x e^{x+\frac{1}{x}}+C$

Naman Kumar
Naman Kumar
Numerade Educator
08:12

Problem 76

If $\int f(x) d x=\psi(x)$, then $\int x^{5} f\left(x^{3}\right) d x$ is equal to [2013 JEE Main]
(a) $\frac{1}{3}\left[x^{3} \psi\left(x^{3}\right)-\int x^{2} \psi\left(x^{3}\right) d x\right]+C$
(b) $\frac{1}{3} x^{3} \psi\left(x^{3}\right)-3 \int x^{3} \psi\left(x^{3}\right) d x+C$
(c) $\frac{1}{3} x^{3} \psi\left(x^{3}\right)-\int x^{2} \psi\left(x^{3}\right) d x+C$
(d) $\frac{1}{3}\left[x^{3} \psi\left(x^{3}\right)-\int x^{3} \psi\left(x^{3}\right) d x\right]+C$

Sandip Ranjan
Sandip Ranjan
Numerade Educator
11:50

Problem 77

If the integral $\int \frac{5 \tan x}{\tan x-2} d x=x+a \log$
$|\sin x-2 \cos x|+k$, then $a$ is equal to
(a) $-1$
(b) $-2$
(c) 1
(d) 2

Sandip Ranjan
Sandip Ranjan
Numerade Educator
04:01

Problem 78

$$
\text { The value of } \sqrt{2} \int \frac{\sin x d x}{\sin \left(x-\frac{\pi}{4}\right)} \text { is }
$$

Kumareshwaran Rathinasabapathy
Kumareshwaran Rathinasabapathy
Numerade Educator