Techniques of Integration

7.3 Assess Your Understanding
Printed Page 493

Concepts and Vocabulary

True or False To find $\int \sqrt{a^{2}-x^{2}}\,dx$ , the substitution $x=a\sin \theta$ , $-\dfrac{\pi }{2} \le \theta \le \dfrac{ \pi }{2}$ , can be used.

True

Multiple Choice To find $\int \sqrt{x^{2}+16}\,dx$ , use the substitution $x=$ [(a) $4\sin \theta ,$ (b) $\tan \theta ,$ (c) $4\sec \theta ,$ (d) $4\tan \theta$ ].

(d)

Multiple Choice To find $\int \sqrt{x^{2}-9} \,dx$ , use the substitution $x=$ [(a) $\sec \theta ,$ (b) $3\sin \theta ,$ (c) $3\sec \theta ,$ (d) $3\tan \theta$ ].

(c)

Multiple Choice To find $\int \sqrt{25-4x^{2}}\,dx$ , use the substitution $x= \bigg[$ (a) $\dfrac{5}{2}\tan \theta ,$ (b) $\dfrac{5}{2} \sin \theta ,$ (c) $\dfrac{2}{5}\sin \theta ,$ (d) $\dfrac{2}{5}\sec \theta \bigg]$ .

(b)

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Skill Building

In Problems 5–14, find each integral. Each of these integrals contains a term of the form $\sqrt{a^{2}-x^{2}}$ .

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

$\dfrac{x}{2}\sqrt{4-x^2}+2\,\sin^{-1}\dfrac{x}{2}+C$

$\int \sqrt{16-x^{2}} dx$

$\int \dfrac{x^{2}}{\sqrt{16-x^{2}}} dx$

$-\dfrac{x}{2}\sqrt{16-x^2}+8\,\sin^{-1}\dfrac{x}{4} + C$

$\int \dfrac{x^{2}}{\sqrt{36-x^{2}}} dx$

$\int \dfrac{\sqrt{4-x^{2}}}{x^{2}} dx$

$-\dfrac{1}{x}\sqrt{4-x^2}-\sin^{-1}\dfrac{x}{2}+C$

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

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

$\dfrac{x(x^2-2)}{4}\sqrt{4-x^2}+2\,\sin^{-1}\dfrac{x}{2}+C$

$\int x^{2}\sqrt{1-16x^{2}} dx$

$\int \dfrac{dx}{(4-x^{2})^{3/2}}$

$\dfrac{1}{4} \dfrac{x}{\sqrt{4-x^2}}+C$

$\int \dfrac{dx}{(1-x^{2})^{3/2}}$

In Problems 15–26, find each integral. Each of these integrals contains a term of the form $\sqrt{x^{2}+a^{2}}.$

$\int \sqrt{4+x^{2}} dx$

$\dfrac{1}{2}x\sqrt{4+x^2} + 2\ln\left|\dfrac{\sqrt{4+x^2}+x}{2}\right|+C$

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

$\int \dfrac{dx}{\sqrt{x^{2}+16}}$

$\ln\left| \dfrac{\sqrt{x^2+16}+x}{4}\right|+C$

$\int \dfrac{dx}{\sqrt{x^{2}+25}}$

$\int {\sqrt{1+9x^{2}}} dx$

$\dfrac{x}{2}\sqrt{1+9x^2}+\dfrac{1}{6}\ln\left| \sqrt{1+9x^2}+3x \right|+C$

$\int {\sqrt{9+4x^{2}}} dx$

$\int \dfrac{x^{2}}{\sqrt{4+9x^{2}}} dx$

$\dfrac{1}{18}x\sqrt{4+9x^2} - \dfrac{2}{27}\ln \left| \dfrac{\sqrt{4 + 9x^2} + 3x}{2} \right| +C$

$\int \dfrac{x^{2}}{\sqrt{x^{2}+16}} dx$

$\int \dfrac{dx}{x^{2}\sqrt{x^{2}+4}}$

$-\dfrac{\sqrt{x^2+4}}{4x}+C$

$\int \dfrac{dx}{x^{2}\sqrt{4x^{2}+1}}$

$\int \dfrac{dx}{(x^{2}+4)^{3/2}}$

$\dfrac{x}{4\sqrt{x^2+4}}+C$

$\int \dfrac{dx}{(x^{2}+1)^{3/2}}$

In Problems 27–36, find each integral. Each of these integrals contains a term of the form $\sqrt{x^{2}-a^{2}}.$

$\int \dfrac{x^{2}}{\sqrt{x^{2}-25}} dx$

$\dfrac{x}{2}\sqrt{x^2-25}+\dfrac{25}{2}\ln \left| \dfrac{x+\sqrt{x^2-25}}{5} \right| + C$

$\int \dfrac{x^{2}}{\sqrt{x^{2}-16}} dx$

$\int \dfrac{\sqrt{x^{2}-1}}{x} dx$

$\sqrt{x^2-1}-\sec^{-1}x + C$

$\int \dfrac{\sqrt{x^{2}-1}}{x^{2}} dx$

$\int \dfrac{dx}{x^{2}\sqrt{x^{2}-36}}$

$\dfrac{\sqrt{x^2-36}}{36x}+C$

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

$\int \dfrac{dx}{\sqrt{4x^{2}-9}}$

$\dfrac{1}{2} \ln \left| \dfrac{2x+\sqrt{4x^2-9}}{3} \right| + C$

$\int \dfrac{dx}{\sqrt{9x^{2}-4}}$

$\int \dfrac{dx}{(x^{2}-9)^{3/2}}$

$-\dfrac{1}{9} \dfrac{x}{\sqrt{x^2-9}}+C$

$\int \dfrac{dx}{(25x^{2}-1)^{3/2}}$

In Problems 37–48, find each integral.

$\int \dfrac{x^{2}\,dx}{(x^{2}-9)^{3/2}}$

$-\dfrac{x}{\sqrt{x^2-9}}+\ln \left| \dfrac{x+\sqrt{x^2-9}}{3} \right| +C$

$\int \dfrac{x^{2}\,dx}{(x^{2}-4)^{3/2}}$

$\int \dfrac{x^{2}\,dx}{16+x^{2}}$

$x-4\,\tan^{-1}\dfrac{x}{4}+C$

$\int \dfrac{x^{2}\,dx}{1+16x^{2}}$

$\int \sqrt{4-25x^{2}} dx$

$\dfrac{x}{2}\sqrt{4-25x^2}+\dfrac{2}{5}\sin^{-1}\dfrac{5x}{2}+C$

$\int \sqrt{9-16x^{2}} dx$

$\int \dfrac{dx}{(4-25x^{2})^{3/2}}$

$\dfrac{x}{4\sqrt{4-25x^2}}+C$

$\int \dfrac{dx}{(1-9x^{2})^{3/2}}$

$\int \sqrt{4+25x^{2}} dx$

$\dfrac{1}{2}x \sqrt{4+25x^2} + \dfrac{2}{5} \ln \left| \dfrac{\sqrt{4+25x^2} + 5x}{2} \right| +C$

$\int \sqrt{9+16x^{2}} dx$

$\int \dfrac{dx}{x^{3}\sqrt{x^{2}-16}}$

$\dfrac{1}{128}\sec^{-1}\dfrac{x}{4} + \dfrac{\sqrt{x^2-16}}{32x^2} +C$

$\int \dfrac{dx}{x^{3}\sqrt{x^{2}-1}}$

In Problems 49–58, find each definite integral.

$\int_{0}^{1}\sqrt{1-x^{2}} dx$

$\dfrac{\pi}{4}$

$\int_{0}^{1/2}\sqrt{1-4x^{2}} dx$

$\int_{0}^{1}\sqrt{1+x^{2}} dx$

$\dfrac{\sqrt{2}+\ln\big(1+\sqrt{2}\big)}{2}$

$\int_{0}^{2}\dfrac{x^{2}}{\sqrt{9+x^{2}}} dx$

$\int_{4}^{5}\dfrac{x^{2}}{\sqrt{x^{2}-9}} dx$

$10-2\sqrt{7}+9\ln3-\dfrac{9}{2}\ln \big(4+\sqrt{7}\big)$

$\int_{1}^{2}\dfrac{x^{2}}{\sqrt{4x^{2}-1}} dx$

$\int_{0}^{2}\dfrac{x^{2}\,dx}{(16-x^{2})^{3/2}}$

$\dfrac{\sqrt{3}}{3} - \dfrac{\pi}{6}$

$\int_{0}^{1}\dfrac{x^{2}\,dx}{(25-x^{2})^{3/2}}$

$\int_{0}^{3}\dfrac{x^{2}\,dx}{9+x^{2}}$

$3-\dfrac{3\pi}{4}$

$\int_{0}^{1}\dfrac{x^{2}}{25+x^{2}} dx$

Applications and Extensions

Area of an Ellipse Find $\int \sqrt{a^{2}-x^{2}} dx$ and use it to find the area enclosed by the ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{ y^{2}}{b^{2}}=1$ .

$\pi ab$

Area of a Semicircle
1. Find ${\int_{0}^{2}{\sqrt{4-x^{2}} \,dx}}$ by interpreting the integral as a certain area and using elementary geometry.
2. Find ${\int_{0}^{2}{\sqrt{4-x^{2}}\,dx}}$ using a trigonometric substitution.
3. Find the area of the semicircle $y=\sqrt{a^{2}-x^{2}},$ $-a \le x \le a$ , using integration.

Average Value Find the average value of the function $f( x) =\dfrac{1}{\sqrt{9-4x^{2}}}$ over the interval [0,1].

$\dfrac{1}{2}\sin^{-1}\dfrac{2}{3}$

Average Value Find the average value of the function $f( x) =\sqrt{x^{2}-4}$ the interval [2,7].

Area Under a Graph Find the area under the graph of $y=\dfrac{x^{3}}{\sqrt{9-x^{2}}}$ from $x=0$ to $x=2$ .

$18-\dfrac{22\sqrt{5}}{3}$

Area Under a Graph Find the area under the graph of $y=x\sqrt{ 16-x^{2}},$ $x \ge 0$ .

495

Area Under a Graph Find the area under the graph of $y=\dfrac{ x^{2}}{\sqrt{x^{2}-1}}$ from $x=3$ to $x=5$ .

$\dfrac{1}{2}\ln\big(2\sqrt{6}+5\big) - \dfrac{1}{2}\ln\big(2\sqrt{2} + 3\big) - 3\sqrt{2} + 5\sqrt{6}$

Hydrostatic Force A round window of radius $2$ meters ( m) is built into the side of a large, fresh-water aquarium tank. If the center of the window is $3 \hbox{m}$ below the water line, find the force due to hydrostatic pressure on the window. (Hint: The mass density of fresh water is ${\small \rho =1000}\hbox{kg}/\hbox{m}^{3}.)$

Area of a Lune A lune is a crescent-shaped area formed when two circles intersect.
1. Find the area of the smaller lune formed by the intersection of the two circles $x^{2}+y^{2}=4$ and $x^{2}+\left( y-2\right) ^{2}=1$ , as shown in the figure.
2. (b) What is the area of the larger lune?

$4\sin^{-1}\dfrac{\sqrt{15}}{8}+\sin^{-1}\dfrac{\sqrt{15}}{4} - \dfrac{\sqrt{15}}{2}$
$\pi - 4\sin^{-1}\dfrac{\sqrt{15}}{8}-\sin^{-1}\dfrac{\sqrt{15}}{4} + \dfrac{\sqrt{15}}{2}$

Area Find the area enclosed by the hyperbola $\dfrac{x^{2}}{9}-\dfrac{y^{2}}{16}=1$ and the line $x=6$ .

Arc Length Find the length of the graph of the parabola $y=5x-x^{2}$ that lies above the $x$ -axis.

$\dfrac{1}{4}\ln\big(\sqrt{26} + 5\big) - \dfrac{1}{4}\ln\big(\sqrt{26}-5\big)+\dfrac{5}{2}\sqrt{26}$

Arc Length Find the length of the graph of $y=\ln x$ from $x= \dfrac{\sqrt{3}}{3}$ to $x=\sqrt{3}$ .

Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region bounded by the graph of $y=\dfrac{1}{x^{2}+4}$ and the $x$ -axis from $x=0$ to $x=1$ about the $x$ -axis. See the figure.

$\dfrac{\pi}{40}+\dfrac{\pi}{16}\tan^{-1}\dfrac{1}{2}$

Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region bounded by the graphs of $y= \dfrac{1}{\sqrt{9-x^{2}}}$ , $y=0$ , $x=0$ , and $x=2$ about the $x$ -axis.

In Problems 73–78, find each integral. (Hint: Begin with a substitution.)

$\int \dfrac{dx}{\sqrt{1-(x-2)^{2}}}$

$\sin^{-1}(x-2)+C$

$\int \sqrt{4-(x+2)^{2}}dx$

$\int \dfrac{dx}{\sqrt{(x-1)^{2}-4}}$

$\ln\left|x-1+\sqrt{(x-1)^2-4}\right|+C$

$\int \dfrac{dx}{(x-2)\sqrt{(x-2)^{2}+9}}$

$\int e^{x}\sqrt{25-e^{2x}}dx$

$\dfrac{25}{2}\sin^{-1}\dfrac{e^x}{5} + \dfrac{1}{2}e^x\sqrt{25-e^{2x}}+C$

$\int e^{x}\sqrt{4+e^{2x}}dx$

In Problems 79 and 80, use integration by parts and then the methods of this section to find each integral.

$\int x\sin ^{-1}x\,dx$

$\dfrac{1}{2}x^{2}\sin^{-1}x - \dfrac{1}{4}\sin^{-1}x + \dfrac{1}{4}x\sqrt{1-x^2} + C$

$\int x\cos ^{-1}x\,dx$

Find $\int \sqrt{x^{2}+a^{2}}\,dx$
1. (a) By using a trigonometric substitution
2. (b) By using substitution with a hyperbolic function

$\dfrac{x}{2}\sqrt{x^2+a^2}+\dfrac{a^2}{2}\ln \left|\dfrac{x+\sqrt{x^2+a^2}}{a}\right|+C$
$\dfrac{x}{2}\sqrt{x^2+a^2}+\dfrac{a^2}{2}\sinh^{-1}\dfrac{x}{a}+C$

In Problems 82–86, use a trigonometric substitution to derive each formula. Assume $a\gt0$ .

$\int \dfrac{dx}{\sqrt{a^{2}-x^{2}}}=\sin ^{-1} \dfrac{x}{a} +C$

$\int \dfrac{dx}{a^{2}+x^{2}}=\dfrac{1}{a}\tan ^{-1} \dfrac{x}{a} +C$

See the Student Solutions Manual.

$\int \dfrac{dx}{x\sqrt{x^{2}-a^{2}}}=\dfrac{1}{a}\sec ^{-1} \dfrac{x}{a} +C$

$\int \dfrac{dx}{\sqrt{x^{2}-a^{2}}}=\ln \left| \dfrac{x+\sqrt{x^2-a^2}}{a} \right| +C$

See the Student Solutions Manual.

$\int \dfrac{dx}{\sqrt{x^{2}+a^{2}}}=\ln \big| x+\sqrt{x^{2}+a^{2}}\big| +C$

Challenge Problems

Find $\int \dfrac{dx}{\sqrt{3x-x^2}}$

$\sin^{-1}\dfrac{2x-3}{3}+C$

Derive the formula $\int \sqrt{x^{2}-a^{2}}dx=\dfrac{1}{2}x \sqrt{x^{2}-a^{2}}-\dfrac{1}{2}a^{2}\ln \vert x+\sqrt{x^{2}-a^{2}}\vert + C, a \gt 0.$

Find $\int \dfrac{\,dx}{\sqrt{x^{2}+a^{2}}}$ , $a\gt0$ , using the substitution $u=\sinh ^{-1} \dfrac{x}{a}$ . Express your answer in logarithmic form.

$\ln\dfrac{x+\sqrt{x^2+a^2}}{a} + C$

Find $\int \dfrac{\sec ^{2}x}{\sqrt{\tan ^{2}x-6\tan x+8}}\,dx$ .

7.3 Assess Your UnderstandingPrinted Page 493

7.3 Assess Your Understanding
Printed Page 493