Integrate the function x e^ x (1+x)^ 2 - Vidyadip

Question

Integrate the function $\frac{x e^{x}}{(1+x)^{2}}$

✓

Answer

$I=\int \frac{x e^{x}}{(1+x)^{2}} d x=\int e^{x}\left\{\frac{x}{(1+x)^{2}}\right\} d x$
= $\int e^{x}\left\{\frac{1+x-1}{(1+x)^{2}}\right\} d x$
= $\int e^{x}\left\{\frac{1}{1+x}-\frac{1}{(1+x)^{2}}\right\} d x$
Now,
Let $\frac{1}{1+x}=f(x) \Rightarrow f^{\prime}(x)=-\frac{1}{(1+x)^{2}}$
We know that,
$\int e^{x}\left\{f(x)+f^{\prime}(x)\right\} d x=e^{x} f(x)+C$
Thus,
$\int \frac{x e^{x}}{(1+x)^{2}} d x=\frac{e^{x}}{1+x}+C$

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 "1 Marks Question" 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

Integrate the functions in Exercises:
$\frac{\text{x}}{\text{e}^{\text{x}^2}}$

If $\vec{\text{a}}=\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}}\text{ and }\vec{\text{b}}=3\hat{\text{i}}+\hat{\text{j}}-5\hat{\text{k}}$ find a unit vector in the direction of $\vec{\text{a}}-\vec{\text{b}}$.

Find the integral: $\int \frac{d x}{x^{2}-6 x+13}$

If P(A) = $\frac{6}{11}$, P(B) = $\frac{5}{11}$ and $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{7}{11}$, find P(B|A)

Examine the continuity of f, where f is defined by $f(x)=\left\{\begin{array}{ll} {\sin x-\cos x,} & {\text { if } x \neq 0} \\ {-1,} & {\text { if } x=0} \end{array}\right.$

Compute the following sums:
$\begin{bmatrix}3&-2\\1&4\end{bmatrix}+\begin{bmatrix}-2&4\\1&3\end{bmatrix}$

Determine the value of ‘k’ for which the following function is continuous at x = 3:
$ \text{f(x)} = \begin{cases} \frac{(\text{x + 3)}^{2} \text{ - } 36}{\text{x - 3}} & , & \text{x}\neq 3 \\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{k}& , & \text{x = 3}\\ \end{cases}$

Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$\frac{\text{dy}}{\text{dx}}+\text{e}^\text{y}=0$

If $\left|\begin{array}{cc} {x} & {2} \\ {18} & {x} \end{array}\right|=\left|\begin{array}{cc} {6} & {2} \\ {18} & {6} \end{array}\right|$ , then $x$ is equal to

Write the number of vectors of unit length perpendicular to both the vectors $\overrightarrow{\text{a}} = 2\hat{\text{i}} + \hat{j} + 2\hat{\text{k}} \text{ and} \overrightarrow{\text{b}} = \hat{\text{j}} + \hat{\text{k}}.$