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

Question

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

✓

Answer

Let $\text{I}=\int\text{e}^{\text{x}}\frac{1+\text{x}}{(2+\text{x})^2}\text{dx}$
$=\int\text{e}^{\text{x}}\Big(\frac{\text{x}+2-1}{(2+\text{x})^2}\Big)\text{dx}$
$=\int\text{e}^{\text{x}}\bigg\{\frac{1}{\text{x}+2}-\frac{1}{(\text{x}+2)^2}\bigg\}\text{dx}$
$=\int\text{e}^{\text{x}}\frac{1}{\text{x}+2}\text{dx}-\int\text{e}^{\text{x}}\frac{1}{(\text{x}+2)^2}\text{dx}$
Integrating by parts
$=\text{e}^{\text{x}}\frac{1}{\text{x}+2}-\int\text{e}^{\text{x}}\Big(\frac{\text{d}}{\text{dx}}\Big(\frac{1}{\text{x}+2}\Big)\Big)\text{dx}-\int\text{e}^{\text{x}}\frac{1}{(\text{x}+2)^2}\text{dx}$
$=\text{e}^{\text{x}}\frac{1}{\text{x}+2}+\int\text{e}^{\text{x}}\frac{1}{(\text{x}+2)^2}\text{dx}-\int\text{e}^{\text{x}}\frac{1}{(\text{x}+2)^2}\text{dx}$
$=\frac{\text{e}^{\text{x}}}{\text{x}+2}+\text{C}$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Prove that: $\begin{vmatrix}(b+c)^2&a^2&\text{bc}\\(c+a)^2&b^2&\text{ca}\\(a+b)^2&c^2&\text{ab}\end{vmatrix}$ $=(a-b)(b-c)(c-b)(a+b+c)(a^2+b^2+c^2)$

The slope of the tangent at each point of a curve is equal to the sum of the coordinates of the point. Find the curve that passes through the origin.

If A and B are two events such that $2\text{P(A)}=\text{P(B)}=\frac{5}{13}$ and $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)=\frac{2}{5}$ find $\text{P}(\text{A}\cap\text{B}).$

If the curves $a x^2+b y^2=1$ and $a^{\prime} x^2+b^{\prime} y^2=1$, intersect orthogonally, then prove that

$\frac{1}{a}-\frac{1}{b}=\frac{1}{a^{\prime}}-\frac{1}{b^{\prime}}$

Integrate the following w. r. t. x:

$\frac{12 x+3}{6 x^2+13 x-63}$

If $\text{A}=\begin{bmatrix}1&1\\0&1\end{bmatrix},$ show that $\text{A}^2=\begin{bmatrix}1&2\\0&1\end{bmatrix}$ and $\text{A}^3=\begin{bmatrix}1&3\\0&1\end{bmatrix}.$

Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\text{x}^{\sin\text{x}}+\big(\sin\text{x}\big)^\text{x}$

Find the slopes of the tangent and the normal to the following curves at the indicated points:
$\text{x}=\text{a}(\theta-\sin\theta),\text{y}=\text{a}(1-\cos\theta)\ \text{at}\ \theta=\frac{-\pi}{2}$

Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=(\tan\text{x})^{\cot\text{x}}+(\cot\text{x})^{\tan\text{x}}$

Assume that a rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop.