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Applied Calculus For Business, Economics, and Finance

Warren B. Gordon, Walter O. Wang, April Allen Materowski

Chapter 5

Integration and its Applications

Educators


Problem 1

Evaluate the given integral and check your answer.
$$\int 3 d x$$

Gregory H.
Numerade Educator

Problem 2

Evaluate the given integral and check your answer.
$$\int x^{3} d x$$

Gregory H.
Numerade Educator

Problem 3

Evaluate the given integral and check your answer.
$$\int \frac{1}{x^{3}} d x$$

Gregory H.
Numerade Educator

Problem 4

Evaluate the given integral and check your answer.
$$\int \sqrt{x} d x$$

Gregory H.
Numerade Educator

Problem 5

Evaluate the given integral and check your answer.
$$\int \sqrt{t} d t$$

Gregory H.
Numerade Educator

Problem 6

Evaluate the given integral and check your answer.
$$\int \sqrt[3]{x} d x$$

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Numerade Educator

Problem 7

Evaluate the given integral and check your answer.
$$\int \sqrt[4]{t} d t$$

Gregory H.
Numerade Educator

Problem 8

Evaluate the given integral and check your answer.
$$\int \frac{1}{\sqrt{x}} d x$$

Gregory H.
Numerade Educator

Problem 9

Evaluate the given integral and check your answer.
$$\int x^{3 / 4} d x$$

Gregory H.
Numerade Educator

Problem 10

Evaluate the given integral and check your answer.
$$\int \frac{1}{w^{5 n}} d w$$

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Numerade Educator

Problem 11

Evaluate the given integral and check your answer.
$$\int \sqrt{s^{3}} d s$$

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Numerade Educator

Problem 12

Evaluate the given integral and check your answer.
$$\int \sqrt[4]{x^{5}} d x$$

Gregory H.
Numerade Educator

Problem 13

Evaluate the given integral and check your answer.
$$\int \frac{1}{\sqrt[3]{x^{2}}} d x$$

Gregory H.
Numerade Educator

Problem 14

Evaluate the given integral and check your answer.
$$\int 2 t^{7} d t$$

Gregory H.
Numerade Educator

Problem 15

Evaluate the given integral and check your answer.
$$\int \frac{1}{2 w^{3}} d w$$

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Numerade Educator

Problem 16

Evaluate the given integral and check your answer.
$$\int\left(2 r^{3}-\frac{3}{r^{2}}+4\right) d r$$

Gregory H.
Numerade Educator

Problem 17

Evaluate the given integral and check your answer.
$$\int\left(5 x^{2}+2 \sqrt{x}-3\right) d x$$

Gregory H.
Numerade Educator

Problem 18

Evaluate the given integral and check your answer.
$$\int(3 x)^{3} d x$$

Gregory H.
Numerade Educator

Problem 19

Evaluate the given integral and check your answer.
$$\int(3 x)^{3} d x$$

Check back soon!

Problem 20

Evaluate the given integral and check your answer.
$$\int\left(t^{2}+1\right)^{2} d t$$

Gregory H.
Numerade Educator

Problem 21

Evaluate the given integral and check your answer.
$$\int \sqrt{2 x} d x$$

Gregory H.
Numerade Educator

Problem 22

Evaluate the given integral and check your answer.
$$\int \frac{4 x^{3}-7 x+2}{x^{5}} d x$$

Gregory H.
Numerade Educator

Problem 23

Evaluate the given integral and check your answer.
$$\int \frac{5 t^{7}-2 t^{4}+3}{2 t^{3}} d t$$

Gregory H.
Numerade Educator

Problem 24

Evaluate the given integral and check your answer.
$$\int \frac{4}{x} d x$$

Gregory H.
Numerade Educator

Problem 25

Evaluate the given integral and check your answer.
$$\int \frac{2}{x} d x$$

Gregory H.
Numerade Educator

Problem 26

Evaluate the given integral and check your answer.
$$\int 7 e^{x} d x$$

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Numerade Educator

Problem 27

Evaluate the given integral and check your answer.
$$\int 2 e^{t} d t$$

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Numerade Educator

Problem 28

Evaluate the given integral and check your answer.
$$\int\left(\frac{5}{x}-2 e^{x}+7\right) d x$$

Gregory H.
Numerade Educator

Problem 29

Evaluate the given integral and check your answer.
$$\int\left(2 e^{t}-3 t^{4}+\frac{7}{t}-12\right) d t$$

Gregory H.
Numerade Educator

Problem 30

Evaluate the given integral and check your answer.
$$\int\left(4 x^{3}-9 e^{x}+\frac{8}{x}-5\right) d x$$

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Numerade Educator

Problem 31

Compute $\frac{d}{d x}\left(\frac{1}{a} e^{a x}\right),$ where the constant $a \neq 0 .$ Use this result to prove that $\int e^{a x} d x=\frac{1}{a} e^{a x}+C.$

Gregory H.
Numerade Educator

Problem 32

Using the result of Exercise $31,$ evaluate (a) $\int e^{2 x} d x,$ (b) $\int e^{-x} d x$ (c) $\int \frac{1}{e^{4 x}} d x.$

Gregory H.
Numerade Educator

Problem 33

(a) Compute $\frac{d}{d x}(x \ln x-x)$. (b) What function can you now antidifferentiate?

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Numerade Educator

Problem 34

(a) Compute $\frac{d}{d x}\left(\frac{1}{b} \ln (a+b x)\right) ;$ use this to determine $\int \frac{1}{a+b x} d x.$

Gregory H.
Numerade Educator

Problem 35

We have that $\int \frac{1}{x} d x=\ln x+c,$ but it also follows from $\frac{d}{d x}(\ln a x)=\frac{1}{x},$ that we have $\int \frac{1}{x} d x=\ln a x+C .$ Is there a problem with these two different results for the same integrand? Explain.

Gregory H.
Numerade Educator