Evaluate the integral in Exercise: ^ 2 _ 1 (1 x -1 2x^ 2 )e^ 2x d… - Vidyadip

Question

Evaluate the integral in Exercise:
$\int^{2}_{1}\bigg(\frac{1}{\text{x}}-\frac{1}{2\text{x}^{2}}\bigg)\text{e}^{2\text{x}}\text{dx}$

✓

Answer

$\text{Let}$
$\text{I}=\int_{1}^{2}\bigg(\frac{1}{\text{x}}-\frac{1}{2\text{x}^{2}}\bigg)\text{e}^{2\text{x}}\text{dx}$
$=\int^{2}_{1}\frac{1}{\text{x}}\text{e}^{2\text{x}}\text{dx}+\int\frac{-1}{2\text{x}^{2}}\text{e}^{2\text{x}}\text{dx}$
$=\bigg[\frac{1}{\text{x}}\frac{\text{e}^{2\text{x}}}{2}\bigg]^{2}_{1}-\int^{2}_{1}\frac{-1}{\text{x}^{2}}\frac{\text{e}^{2\text{x}}}{2}\text{dx}+\int^{2}_{1}\frac{-1}{2\text{x}^{2}}\text{e}^{2\text{x}}\text{dx}$ [integrating by parts]
$=\frac{1}{2}\bigg[\frac{\text{e}^{2\text{x}}}{\text{x}}\bigg]^{2}_{1}-\int^{2}_{1}\frac{-1}{2\text{x}^{2}}{\text{e}^{2\text{x}}}{}\text{dx}+\int^{2}_{1}\frac{-1}{2\text{x}^{2}}\text{e}^{2\text{x}}\text{dx}$
$=\frac{1}{2}\bigg[\frac{\text{e}^{4}}{2}-\frac{\text{e}^{2}}{1}\bigg]=\frac{1}{4}\big(\text{e}^{4}-2\text{e}^{2}\big)=\frac{1}{4}\big(\text{e}^{2}-2\big)$

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 "3 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

Given the matrices
$\text{A}=\begin{bmatrix}2&1&1\\3&-1&0\\0&2&4\end{bmatrix},\text{B}=\begin{bmatrix}9&7&-1\\3&5&4\\2&1&6\end{bmatrix}$ and $\text{C}=\begin{bmatrix}2&-4&3\\1&-1&0\\9&4&5\end{bmatrix}$ Verify that (A + B) + C = A + (B + C).

If $y=\log \left[x+\sqrt{a^2+x^2}\right]$, then prove that $\left(a^2+x^2\right) \frac{d^2 y}{d x^2}+x \frac{d y}{d x}=0$.

There are two bags I and II. Bag I contains 2 white and 3 red balls and Bag II contains 4 white and 5 red balls. One ball is drawn at random from one of the bags and is found to be red. Find the probability that it was drawn from bag II.

If $\text A = \begin{bmatrix}3&-2\\4&-2\end{bmatrix} \text {and I} = \begin{bmatrix}1&0\\0&1\end{bmatrix},$ find k so that $A^2 = kA - 2I$.

Express the following matrix as the sum of a symmetric and a skew symmetric matrix:$ \begin{bmatrix} 1 & 3 & 5 \\ - 6 & 8 & 3 \\ - 4 & 6 & 5 \end{bmatrix} $

Let $A = {1, 2, 3},$ and let $R_1 = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}$. Find whether or not the relations $R_1$ on $A$ is:

Reflexive.
Symmetric.
Transitive.

Show that the vector $\hat i + \hat j + \hat k$ is equally inclined to the axes OX, OY and OZ.

Differentiate the function with respect to x : $2 \sqrt{\cot \left(x^{2}\right)}$

Evaluate the following integrals:
$\int\log(\text{x}+1)\text{dx}$

Find $\frac{1}{2}\left(\mathrm{~A}+\mathrm{A}^{\prime}\right)$ and $\frac{1}{2}\left(\mathrm{~A}-\mathrm{A}^{\prime}\right)$ when $A=\left[\begin{array}{ccc}0 & a & b \\ -a & 0 & c \\ -b & -c & 0\end{array}\right]$.