Evaluate the following integrals: (x ^ -1 x) (1+x^2)^ 3 2 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{(\text{x}\tan^{-1}\text{x})}{(1+\text{x}^2)^{\frac{3}{2}}}\text{dx}$

✓

Answer

Let $\text{I}=\int\frac{\text{x}\tan^{-1}\text{x}}{(1+\text{x}^2)^{\frac{3}{2}}}\text{dx}$
putting $\text{x}=\tan\theta$
$\Rightarrow\text{dx}=\sec^2 \theta\text{d}\theta$
$\&\theta\tan^{-1}\text{x}$
$\therefore\text{I}=\int\frac{(\tan\theta).\theta\sec^2\theta\text{d}\theta}{\big(1+\tan^2\theta\big)^{\frac{3}{2}}}$
$=\int\frac{\theta.\tan\theta\sec^2\theta\text{d}\theta}{(\sec^2\theta)^{\frac{3}{2}}}$
$=\int\frac{\theta\tan\theta.\sec^2\theta\text{d}\theta}{\sec^3\theta}$
$=\int\frac{\theta.\tan\theta}{\sec\theta}\text{d}\theta$
$=\int\theta.\sin\theta\text{d}\theta$
$=\theta\int\sin\theta\text{d}\theta-\int\big\{\frac{\text{d}}{\text{d}\theta}(\theta)\int\sin\text{d}\theta\big\}\text{d}\theta$
$=\theta(-\cos\theta)-\int1.(-\cos\theta)\text{d}\theta$
$=-\theta\cos\theta+\sin\theta+\text{C}$
$=\frac{-\theta}{\sec\theta}+\frac{1}{\text{cosec}\theta}+\text{C}$
$=\frac{-\theta}{\sqrt{1+\tan^2\theta}}+\frac{1}{\sqrt{1+\cot^2}\theta}+\text{C}$
$=\frac{-\theta}{\sqrt{1+\tan^2\theta}}+\frac{\tan\theta}{\sqrt{\tan^2\theta+1}}+\text{C}$
$=\frac{-\tan^{-1}\text{x}}{\sqrt{1+\text{x}^2}}+\frac{\text{x}}{\sqrt{\text{x}^2+1}}+\text{C}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the equation of the normal to the curve x² + 2y² - 4x - 6y + 8 = 0 at the point whose abscissa is 2.

Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection,

f'(x) = x⁴- 62x² + 9x + 15

Evaluate the following integrals:

$\int\frac{\cos\text{x}-\sin\text{x}}{\sqrt{8-\sin2\text{x}}}\text{ dx}$

Differentiate the following functions with respect to x:
$\frac{\text{x}^2(1-\text{x}^2)}{\cos2\text{x}}$

Find the area lying above the x-axis and under the paraola y² = 4x - x².

If A and B are independent events such that P(A) = p, P(B) = 2p and P(Exactly one of A and B occurs) $=\frac{5}{9},$ then find the value or p.

A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines and each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at Rs. 7 profit and that of B at a profit of Rs. 4. Find the production level per day for maximum profit graphically.

Find the matrix X satisfying the matrix equation.

$\text{X}\begin{bmatrix}5 & 3 \\-1 & -2 \end{bmatrix}\begin{bmatrix}14 & 7 \\7 & 7 \end{bmatrix}$

Using integration, find the area of the region bounded by the triangle whose vertices are (2, –2), (4, 5) and (6, 2).

Integrate the rational function $\frac{3 x+5}{x^{3}-x^{2}-x+1}$