Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluate the following integrals:$\int\text{e}^{\text{x}}\frac{(1-\text{x})^2}{(1+\text{x}^2)^2}\text{dx}$
✓

Answer

Let $\text{I}=\int\text{e}^{\text{x}}\bigg[\frac{(1-\text{x})^2}{(1+\text{x}^2)^2}\bigg]\text{dx}$
$=\int\text{e}^\text{x}\bigg[\frac{1+\text{x}^2-2\text{x}}{(1+\text{x}^2)^2}\bigg]\text{dx}$
$=\int\text{e}^{\text{x}}\bigg[\frac{1}{1+\text{x}^2}-\frac{2\text{x}}{(1+\text{x}^2)^2}\bigg]\text{dx}$
Here, $\text{f(x)}=\frac{1}{1+\text{x}^2}$
$\Rightarrow\text{f}'(\text{x})=\frac{-2\text{x}}{(1+\text{x}^2)^2}$
Put $\text{e}^{\text{x}}\text{f(x)}=\text{t}$
$\Rightarrow\text{e}^{\text{x}}\frac{1}{1+\text{x}^2}=\text{t}$
Diff. both sides w.r.t w
$\text{e}^{\text{x}}\frac{1}{1+\text{x}^2}+\text{e}^{\text{x}}\frac{-1}{(1+\text{x}^2)^2}2\text{x}=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow\text{e}^{\text{x}}\bigg[\frac{1}{1+\text{x}^2}-\frac{2\text{x}}{(1+\text{x}^2)^2}\bigg]\text{dx = dt}$
$\therefore\int\text{e}^{\text{x}}\bigg[\frac{1}{1+\text{x}^2}-\frac{2\text{x}}{(1+\text{x}^2)^2}\bigg]\text{dx}=\int\text{dt}$
$\Rightarrow\text{I}=\text{t}+\text{C}$
$=\frac{\text{e}^{\text{x}}}{1+\text{x}^2}+\text{C}$

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 Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If the area bounded by the parabola $y^2 = 4ax$ and the line y = mx is $\frac{\text{a}^2}{12}\text{ sq.}$ units, then using integration, find the value of m.
→
Evaluate the following integrals:
$\int^\limits{\frac{\pi}{2}}_{0}\frac{\tan\text{x}}{1+\text{m}^2\tan^2\text{x}}\text{ dx}$
→
Find the area of the region bounded by the paraola $y^2 = 4ax$ and line $x = a.$
→
The sum of the surface areas of a sphere and a cube is given. Show that when the sum of their volumes is least, the diameter of the sphere is equal to the edge of the cube.
→
Find the equation of the tangent to the curve $\text{y}=\sqrt{3}\text{x}-2$ which is parallel to the 4x − 2y + 5 = 0.
→
Verify the Rolle’s theorem for each of the functions:
$\text{f(x)}=\log(\text{x}^2+2)-\log3\text{ in }[-1,1].$
→
Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix} 1 & 1 & 2 \\ 3 & 1 & 1 \\ 2 & 3 & 1 \end{bmatrix}$
→
A unit vector $\vec{\text{a}}$ makes angles $\frac{\pi}{4}$ and $\frac{\pi}{3}$ with $\hat{\text{i}}$ and $\hat{\text{j}}$ respectively and an acute angle $\theta$ with $\hat{\text{k}}$. find the angle $\theta$ and components of $\vec{\text{a}}$ .
→
Using matrices, solve the following system of linear equations:
$x – y + 2z = 7.$
$3x + 4y - 5z = –5.$
$2x – y + 3z = 12.$
→
If $\text{A}=\begin{bmatrix}\text{a}&\text{b}\\0&1\end{bmatrix},$ prove that $\text{A}^\text{n}=\begin{bmatrix}\text{a}^\text{n}&\text{b}\Big(\frac{\text{a}^\text{n}-1}{\text{a}-1}\Big)\\0&1\end{bmatrix}$ for every positive integer n.
→