Q 68. Solve the integral.鈭玿2聽tan-1x... [FREE SOLUTION]

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Chapter 5: Q 68. (page 429)

Solve the integral.
$鈭� x^{2} \tan^{- 1} (x) d x$

Short Answer

Expert verified

The solution is $\frac{x^{3}}{3} \tan^{- 1} (x) - \frac{x^{2}}{6} + \frac{1}{6} \log |x^{2} + 1| + C$ .

Step by step solution

Step 1. Given information.

The given integral is $鈭� x^{2} \tan^{- 1} (x) d x$ .

Step 2. Solve the integral by using the integration by parts method.

$鈭� x^{2} \tan^{- 1} (x) d x = \tan^{- 1} x 鈭� x^{2} d x - 鈭� [\frac{d}{d x} (\tan^{- 1} x) 鈭� x^{2} d x] = \tan^{- 1} x 路 \frac{x^{3}}{3} - 鈭� \frac{1}{1 + x^{2}} 路 \frac{x^{3}}{3} d x = \frac{x^{3}}{3} \tan^{- 1} x - \frac{1}{3} 鈭� (x - \frac{x}{1 + x^{2}}) d x = \frac{x^{3}}{3} \tan^{- 1} x - \frac{1}{3} 鈭� x d x - \frac{1}{3} 鈭� \frac{x}{1 + x^{2}} d x = \frac{x^{3}}{3} \tan^{- 1} x - \frac{1}{3} 路 \frac{x^{2}}{2} - \frac{1}{3} I 鈰� 鈰� 鈰� 鈰� 鈰� 鈰� (1)$

Step 3. Solve for I.

$I = 鈭� \frac{x}{1 + x^{2}} d x$

Substitute $1 + x^{2} = t$ .

$\begin{array}{rcl} 2 x d x & = & d t \\ x d x & = & \frac{d t}{2} \end{array}$

role="math" localid="1649310832867" $\begin{array}{rcl} I & = & \frac{1}{2} 鈭� \frac{d t}{t} \\ = & \frac{1}{2} \log (t) + C \\ = & \frac{1}{2} \log |x^{2} + 1| + C \end{array}$

Step 4. Simplified answer.

Substitute the value of $I$ in equation (1).

$\begin{array}{rcl} 鈭� x^{2} \tan^{- 1} x d x & = & \frac{x^{3}}{3} \tan^{- 1} x - \frac{x^{2}}{6} + \frac{1}{3} I \\ = & \frac{x^{3}}{3} \tan^{- 1} x - \frac{x^{2}}{6} + \frac{1}{3} 路 \frac{1}{2} \log |1 + x^{2}| + C \\ = & \frac{x^{3}}{3} \tan^{- 1} x - \frac{x^{2}}{6} + \frac{1}{6} \log |1 + x^{2}| + C \end{array}$

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91影视

Solve the integral.
$鈭� x^{2} \tan^{- 1} (x) d x$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solve the integral by using the integration by parts method.

Step 3. Solve for I.

Step 4. Simplified answer.

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91影视

Solve the integral.鈭�x2tan-1xdx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solve the integral by using the integration by parts method.

Step 3. Solve for I.

Step 4. Simplified answer.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Probability and Statistics

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Study anywhere. Anytime. Across all devices.

Solve the integral.
$鈭� x^{2} \tan^{- 1} (x) d x$