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

Question

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

✓

Answer

Let $\text{I}=\int\frac{1}{(\text{x}^2-1)\sqrt{\text{x}^2+1}}\text{ dx}$
Let $\text{x}=\frac{1}{\text{t}}$
$\text{dx}=-\frac{1}{\text{t}^2}\text{ dt}$
$\therefore\ \text{I}=-\int\frac{\frac{1}{\text{t}^2}\text{ dt}}{\Big(\frac{1}{\text{t}^2}-1\Big)\sqrt{\Big(\frac{1}{\text{t}^2}+1\Big)}}$
$=-\int\frac{\text{t dt}}{(1-\text{t}^2)\sqrt{1+\text{t}^2}}$
Let $1+\text{t}^2=\text{u}^2$
$2\text{tdt}=2\text{udt}$
$\text{I}=\int\frac{\text{udu}}{(\text{u}^2-2)\text{u}}$
$=\int\frac{\text{du}}{\text{u}^2-2}$
$\therefore\ \text{I}=\frac{1}{2\sqrt{2}}\log\bigg|\frac{\text{u}-\sqrt{2}}{\text{u}+\sqrt{2}}\bigg|+\text{C}$
$=\frac{1}{2\sqrt{2}}\log\bigg|\frac{\sqrt{1+\text{t}^2}-\sqrt{2}}{\sqrt{1+\text{t}^2}+\sqrt{2}}\bigg|+\text{C}$
Hence,
$\text{I}=-\frac{1}{2\sqrt{2}}\log\bigg|\frac{\sqrt{2}\text{x}+\sqrt{\text{x}^2+1}}{\sqrt{2}\text{x}-\sqrt{\text{x}^2+1}}\bigg|+\text{C}$

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 "5 Marks Questions" from INTEGRALS→INTEGRALS — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the direction cosines of the line $\frac{\text{x}+2}{2}=\frac{2\text{y}-7}{6}=\frac{5-\text{z}}{6}.$ Also, find the vector equation of the line through the point A(-1, 2, 3) and parallel to the given line.

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
$\text{f}(\text{x})=\tan^{-1}\text{x}\text{ on }[0,1]$

If $\text{x}\sin(\text{a}+\text{y})+\sin\text{a}\cos(\text{a}+\text{y})=0,$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{\sin^2(\text{a}+\text{y})}{\sin\text{a}}$

If $\cos^{-1}\frac{\text{x}}{2}+\cos^{-1}\frac{\text{y}}{3}=\alpha,$ then prove that $9\text{x}^2-12\text{xy}\cos\alpha+4\text{y}^2=36\sin^2\alpha.$

Differentiate $\tan^{-1}\Big(\frac{\sqrt{1+\text{x}^2}-1}{\text{x}}\Big)$ with respect to $\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big),$ if $-1<\text{x}<1,\text{x}\neq0.$

Solve the following system of equations by matrix method: $6x - 12y + 25z = 4 , 4x + 15y - 20z = 3 , 2x + 18y + 15z = 10$

Find the coordinates of the foot of the perpendicular and the perpendicular distance of the point P(3, 2, 1) from the plane 2x – y + z + 1 = 0. Find also, the image of the point in the plane.

Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{bmatrix}$

If $\text{A}=\begin{bmatrix}1&-1&0\\ 2&3&4\\ 0&1&2\end{bmatrix}\text{and }\text{B}=\begin{bmatrix}2&2&-4\\ -4&2&-4\\ 2&-1&5\end{bmatrix}$ are two square matrices, find AB and hence solve the system of linear equations:
x - y = 3, 2x + 3y + 4z = 17, y + 2z = 7

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\sin\text{x}|\sin\text{x}|\text{dx}$