Techniques of Integration

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7.7 Assess Your Understanding
Printed Page 523

523

Skill Building

In Problems 1–16, find each integral using the Table of Integrals found at the back of the book.

$\int e^{2x}\cos x\,dx$

$\dfrac{1}{5}e^{2x}(2\cos x + \sin x) + C$

$\int e^{5x+1}\sin (2x+3)\,dx$

$\int x\sqrt{4x+3}\,dx$

$\dfrac{1}{20}(2x-1)(3+4x)^{3/2}+C$

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

$\int (x+1) \sqrt{4x+5}\,dx$

$\dfrac{6x+5}{60}(4x+5)^{3/2}+C$

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

$\int \dfrac{dx}{x\sqrt{4x+6}}$

$\dfrac{\sqrt{6}}{6}\ln\left|\dfrac{\sqrt{6+4x}-\sqrt{6}}{\sqrt{6+4x}+\sqrt{6}}\right|+C$

$\int \dfrac{dx}{x\sqrt{8+x}}$

$\int \dfrac{\sqrt{4x+6}}{x}\,dx$

$2\sqrt{6+4x}+\sqrt{6}\ln \left| \dfrac{\sqrt{6+4x}-\sqrt{6}}{\sqrt{6+4x}+\sqrt{6}} \right|+C$

$\int \dfrac{\sqrt{8+x}}{x^{2}}\,dx$

$\int x^{3}(\ln x)^2\,dx$

$\dfrac{x^{4}(\ln x)^{2}}{4} - \dfrac{x^{4}\ln x}{8} + \dfrac{x^{4}}{32} + C$

$\int x^{3} (\ln x)^{2}\,dx$

$\int \sin ^{-1}(2x)\,dx$

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

$\int \tan ^{-1}(-3x)\,dx$

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

$\dfrac{135}{8}\sin^{-1}\dfrac{1}{3} - \dfrac{9\sqrt{2}}{4}$

$\int _{1}^{e}\dfrac{1}{x^{2}\sqrt{x^{2}+2}}\,dx$

In Problems 17–32:

(a) Redo Problems 1-16 using a CAS.
(b) Compare the result to the answer obtained using a Table of Integrals.
(c) If the results are different, verify that they are equivalent.

$\int e^{2x}\cos x\,dx$

$\dfrac{e^{2x}}{5}(\sin x + 2\cos x)+C$
(b) See Student Solutions Manual.
(c) See Student Solutions Manual.

$\int x\sqrt{4x+3}\,dx$

$\dfrac{2x-1}{20}(4x+3)^{3/2}+C$
(b) See Student Solutions Manual.
(c) See Student Solutions Manual.

$\int (x+1) \sqrt{4x+5}\,dx$

$\dfrac{6x+5}{60}(4x+5)^{3/2}+C$
(b) See Student Solutions Manual.
(c) See Student Solutions Manual.

$\int \dfrac{dx}{x\sqrt{4x+6}}$

$-\dfrac{1}{3}\sqrt{6}\tanh^{-1}\left(\dfrac{\sqrt{4x+6}}{6}\sqrt{6}\right)+C$
(b) See Student Solutions Manual.
(c) See Student Solutions Manual.

$\int \dfrac{\sqrt{4x+6}}{x}\,dx$

$2\sqrt{4x+6}-2\sqrt{6}\tanh^{-1}\sqrt{\dfrac{2x}{3}+1}+C$
(b) See Student Solutions Manual.
(c) See Student Solutions Manual.

$\int x^{3}(\ln x)^2\,dx$

$\dfrac{1}{32}x^{4}(8\ln^{2}x - 4\ln x + 1)+C$
(b) See Student Solutions Manual.
(c) See Student Solutions Manual.

$\int \sin ^{-1}(2x)\,dx$

$\dfrac{1}{2}\sqrt{1-4x^2}+x\sin^{-1}(2x)+C$
(b) See Student Solutions Manual.
(c) See Student Solutions Manual.

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

$-\dfrac{9\sqrt{2}}{4}+\dfrac{135}{8}\sin^{-1}\dfrac{1}{3}$
(b) See Student Solutions Manual.
(c) See Student Solutions Manual.

In Problems 33–38, use a CAS to investigate whether each indefinite integral can be expressed using elementary functions.

$\int \sqrt{1+x^{3}}\,dx$

No

$\int \sqrt{1+\sin x}\,dx$

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

No

$\int \dfrac{\cos x}{x}\,dx$

$\int x\tan x\, dx$

No

$\int \sqrt{1+e^{x}}\,dx$