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## Educators

### Problem 1

Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle.

$\displaystyle \int \frac{dx}{x^2 \sqrt{4 - x^2}}$ $x = 2 \sin \theta$

JH
J H.

### Problem 2

Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle.

$\int \frac{ x^3 }{ \sqrt{x^2 + 4}\ } dx$
$x = 2 \tan \theta$

JH
J H.

### Problem 3

Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle.

$\displaystyle \int \frac{\sqrt{x^2 - 4}}{x}\ dx$ $x = 2 \sec \theta$

JH
J H.

### Problem 4

Evaluate the integral.

$\displaystyle \int \frac{x^2}{\sqrt{9 - x^2}}\ dx$

JH
J H.

### Problem 5

Evaluate the integral.

$\displaystyle \int \frac{\sqrt{x^2 - 1}}{x^4}\ dx$

JH
J H.

### Problem 6

Evaluate the integral.

$\displaystyle \int_0^3 \frac{x}{\sqrt{36 - x^2}}\ dx$

JH
J H.

### Problem 7

Evaluate the integral.
$\int_{0}^{a} \frac{d x}{\left(a^{2}+x^{2}\right)^{3 / 2}}, \quad a>0$

JH
J H.

### Problem 8

Evaluate the integral.

$\displaystyle \int \frac{dt}{t^2 \sqrt{t^2 - 16}}$

JH
J H.

### Problem 9

Evaluate the integral.

$\displaystyle \int_2^3 \frac{dx}{(x^2 - 1)^{\frac{3}{2}}}$

JH
J H.

### Problem 10

Evaluate the integral.

$\displaystyle \int_0^{\frac{2}{3}} \sqrt{4 - 9x^2}\ dx$

JH
J H.

### Problem 11

Evaluate the integral.

$\displaystyle \int_0^{\frac{1}{2}} x \sqrt{1 - 4x^2}\ dx$

JH
J H.

### Problem 12

Evaluate the integral.

$\displaystyle \int_0^2 \frac{dt}{\sqrt{4 + t^2}}$

JH
J H.

### Problem 13

Evaluate the integral.

$\displaystyle \int \frac{\sqrt{x^2 - 9}}{x^3}\ dx$

JH
J H.

### Problem 14

Evaluate the integral.

$\displaystyle \int_0^1 \frac{dx}{(x^2 + 1)^2}$

Carson M.

### Problem 15

Evaluate the integral.

$\displaystyle \int_0^a x^2 \sqrt{a^2 - x^2}\ dx$

JH
J H.

### Problem 16

Evaluate the integral.

$\displaystyle \int_{\frac{\sqrt{2}}{3}}^{\frac{2}{3}} \frac{dx}{x^5 \sqrt{9x^2 - 1}}$

JH
J H.

### Problem 17

Evaluate the integral.

$\displaystyle \int \frac{x}{\sqrt{x^2 - 7}}\ dx$

JH
J H.

### Problem 18

Evaluate the integral.

$\displaystyle \int \frac{dx}{[(ax)^2 - b^2]^{\frac{3}{2}}}$

JH
J H.

### Problem 19

Evaluate the integral.

$\displaystyle \int \frac{\sqrt{1 + x^2}}{x}\ dx$

JH
J H.

### Problem 20

Evaluate the integral.

$\displaystyle \int \frac{x}{\sqrt{1 + x^2}}\ dx$

JH
J H.

### Problem 21

Evaluate the integral.

$\displaystyle \int_0^{0.6} \frac{x^2}{\sqrt{9 - 25x^2}}\ dx$

JH
J H.

### Problem 22

Evaluate the integral.

$\displaystyle \int_0^1 \sqrt{x^2 + 1}\ dx$

JH
J H.

### Problem 23

Evaluate the integral.

$\displaystyle \int \frac{dx}{\sqrt{x^2 + 2x + 5}}$

JH
J H.

### Problem 24

Evaluate the integral.

$\displaystyle \int_0^1 \sqrt{x - x^2}\ dx$

JH
J H.

### Problem 25

Evaluate the integral.

$\displaystyle \int x^2 \sqrt{3 + 2x - x^2}\ dx$

JH
J H.

### Problem 26

Evaluate the integral.

$\displaystyle \int \frac{x^2}{(3 + 4x - 4x^2)^{\frac{3}{2}}}\ dx$

JH
J H.

### Problem 27

Evaluate the integral.

$\displaystyle \int \sqrt{x^2 + 2x}\ dx$

JH
J H.

### Problem 28

Evaluate the integral.

$\displaystyle \int \frac{x^2 + 1}{(x^2 - 2x +2)^2}\ dx$

JH
J H.

### Problem 29

Evaluate the integral.

$\displaystyle \int x \sqrt{1 - x^4}\ dx$

JH
J H.

### Problem 30

Evaluate the integral.

$\displaystyle \int_0^{\frac{\pi}{2}} \frac{\cos t}{\sqrt{1 + \sin^2 t}}\ dt$

JH
J H.

### Problem 31

(a) Use trigonometric substitution to show that
$$\displaystyle \int \frac{dx}{\sqrt{x^2 + a^2}} = \ln (x + \sqrt{x^2 + a^2}) + C$$
(b) Use the hyperbolic substitution $x = a \sinh t$ to show that
$$\displaystyle \int \frac{dx}{\sqrt{x^2 + a^2}} = \sinh^{-1} \left (\frac{x}{a} \right) + C$$
These formulas are connected by Formula 3.11.3.

JH
J H.

### Problem 32

Evaluate
$$\displaystyle \int \frac{x^2}{(x^2 + a^2)^{\frac{3}{2}}}\ dx$$
(a) by trigonometric substitution.
(b) by the hyperbolic substitution $x = a \sinh t$.

JH
J H.

### Problem 33

Find the average value of $f(x) = \frac{\sqrt{x^2 -1}}{x}$ , $1 \le x \le 7$.

JH
J H.

### Problem 34

Find the area of the region bounded by the hyperbola $9x^2 - 4y^2 = 36$ and the line $x = 3$.

JH
J H.

### Problem 35

Prove the formula $A = \frac{1}{2} r^2 \theta$ for the area of a sector of a circle with radius $r$ and central angle $\theta$. [Hint: Assume $0 < 0 < \frac{\pi}{2}$ and place the center of the circle at the
origin so it has the equation $x^2 + y^2 = r^2$.Then $A$ is the sum of the area of the triangle $POQ$ and the area of the region $PQR$ in the figure.]

JH
J H.

### Problem 36

Evaluate the integral
$$\displaystyle \int \frac{dx}{x^4 \sqrt{x^2 - 2}}$$
Graph the integrand and its indefinite integral on the same screen and check that your answer is reasonable.

JH
J H.

### Problem 37

Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the curves $y = \frac{9}{(x^2 + 9)}$ , $y = 0$ , $x = 0$ , and $x = 3$.

JH
J H.

### Problem 38

Find the volume of the solid obtained by rotating about the line $x = 1$ the region under the curve $y = x \sqrt{1 - x^2}$ , $0 \le x \le 1$.

JH
J H.

### Problem 39

(a) Use trigonometric substitution to verify that
$$\displaystyle \int_0^x \sqrt{a^2 - t^2}\ dt = \frac{1}{2} a^2 \sin^{-1} \left (\frac{x}{a} \right) + \frac{1}{2} x \sqrt{a^2 - x^2}$$
(b) Use the figure to give trigonometric interpretations of both terms on the right side of the equation in part (a).

JH
J H.

### Problem 40

The parabola $y = \frac{1}{2} x^2$ divides the disk $x^2 + y^2 \le 8$ into two parts. Find the areas of both parts.

JH
J H.

### Problem 41

A torus is generated by rotating the circle $x^2 + (y - R)^2 = r^2$ about the x-axis. Find the volume enclosed by the torus.

JH
J H.

### Problem 42

A charged rod of length $L$ produces an electric field at point $P(a, b)$ given by
$$E(P) = \int_{-a}^{L - a} \frac{\lambda b}{4 \pi \varepsilon_0 (x^2 + b^2)^{\frac{3}{2}}}\ dx$$
where $\lambda$ is the charge density per unit length on the rod and $\varepsilon_0$ is the free space permittivity (see the figure). Evaluate the integral to determine an expression for the electric field $E(P)$.

JH
J H.

### Problem 43

Find the area of the crescent-shaped region (called a $lune$) bounded by arcs of circles with radii $r$ and $R$. (See the figure.)

JH
J H.