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Chapter 5: Q.62 (page 465)

Solve the integral \(\int \frac{1}{(x^2+1)^{5/2}}dx\).

Short Answer

Expert verified

\(\frac{\sin3(\tan^{-1}x)}{12}+\frac{3\sin (\tan^{-1}x)}{4}+C \)

Step by step solution

01

Given Information

Consider the integral \(\int \frac{1}{(x^2+1)^{5/2}}dx\).

02

Calculation

Let \(x=\tan t\) and differentiate it.

\(dx=\sec^2 tdt\)

Substitute into the integral and yields,

\(\begin{align*}I&=\int \frac{\sec^2 tdt}{(\tan^2t+1)^{5/2}}\\&=\int \frac{\sec^2tdt}{\sec^5t}&&[\because \tan^2x+1=\sec^2x]\\&=\int \cos^3tdt&&\left[\because \cos^3x=\frac{\cos 3x+3\cos x}{4}\right]\\&=\frac{1}{4}\int(\cos3t+3\cos t)dt\\&=\frac{\sin3t}{12}+\frac{3\sin t}{4}+C\\&=\frac{\sin3(\tan^{-1}x)}{12}+\frac{3\sin (\tan^{-1}x)}{4}+C \end{align*}\)

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Most popular questions from this chapter

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The substitution x = 2 sec u is a suitable choice for solving鈭�1x2鈭�4dx.

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving鈭�1x2鈭�4dx.

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving鈭�1x2+4dx.

(d) True or False: The substitution x = 2 sin u is a suitable choice for solving鈭�x2+4鈭�5/2dx

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form x2鈭�a2.

(f) True or False: Trigonometric substitution doesn鈥檛 solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.

(g) True or False: When using trigonometric substitution with x=asinu, we must consider the cases x>a and x<-a separately.

(h) True or False: When using trigonometric substitution with x=asecu, we must consider the cases x>a and x<-a separately.

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