Evaluate the following integrals: ^ x (1 x^2 -2 x^3 )dx - Vidyadip

Question

Evaluate the following integrals:
$\int\text{e}^{\text{x}}\Big(\frac{1}{\text{x}^2}-\frac{2}{\text{x}^3}\Big)\text{dx}$

✓

Answer

$\text{I}=\int\text{e}^{\text{x}}\big(\text{x}^{-2}-2\text{x}^{-3}\big)\text{dx}$
$=\int\text{e}^{\text{x}}\text{x}^{-2}\text{dx}-2\int\text{e}^{\text{x}}\text{x}^{-3}\text{dx}$
Integration by parts
$=\text{e}^{\text{x}}\text{x}^{-2}-\int\text{e}^{\text{x}}\Big(\frac{\text{d}}{\text{dx}}\big(\text{x}^{-2}\big)\Big)\text{dx}-2\int\text{e}^{\text{x}}\text{x}^{-3}\text{dx}$
$=\text{e}^{\text{x}}\text{x}^{-2}+2\int\text{e}^{\text{x}}\text{x}^{-3}\text{dx}-2\int\text{e}^{\text{x}}\text{x}^{-3}\text{dx}$
$=\frac{\text{e}^{\text{x}}}{\text{x}^2}+\text{C}$
$\int\text{e}^{\text{x}}\Big(\frac{1}{\text{x}^2}-\frac{2}{\text{x}^3}\Big)\text{dx}=\frac{\text{e}^\text{x}}{\text{x}^2}+\text{C}$

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 "3 Marks" 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

Differentiate the following w.r.t. x:
$\sin\text{x}^2+\sin^2\text{x}+\sin^2(\text{x}^2)$

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\log\Big(\frac{3+5\cos\text{x}}{3+5\sin\text{x}}\Big)\text{dx}$

Let f be a function from R to R, such that f(x) = cos(x + 2). Is f invertible? Justify your answer.

Find the mean μ variance σ² for the following probability distribution:

X	0	1	2	3
P(X)	$\frac{1}{6}$	$\frac{1}{2}$	$\frac{3}{10}$	$\frac{1}{30}$

$\begin{array}{l}\text { Find } \frac{d^2 y}{d x^2}: \\ x=a\left(\cos t+\log \tan \frac{t}{2}\right), y=a \sin t\end{array}$

Find the matrix A such that
$\begin{bmatrix}2&1&3\end{bmatrix}\begin{bmatrix}-1&0&-1\\-1&1&0\\0&1&1\end{bmatrix}\begin{bmatrix}1\\0\\-1\end{bmatrix}=\text{A}$

Prove that the greatest integer function defined by
$\text{f(x)} = [\text{x} ], 0 < \text{x} < 3$
is not differentiable at x = 1 and x = 2.

Solve: $\cos\big(\sin^{-1}\text{x}\big)=\frac{1}{6}$

Evaluate the following integrals:

$\int\sqrt{\text{cosec}\text{x}-1}\text{ dx}$

If in a binomial distribution n = 4 and P(X = 0) $=\frac{16}{81},$ find q.