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 "Solve the Following Question.(3 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

Find the co-ordinates of the points of intersection of the lines represented by $x^2-y^2-2 x+1$ = 0.

The position of a particle is qiven by the function $s(t)=2 t^2+3 t-4$. Find the time $t=c$ in

the interval 0 ≤ f ≤ 4 when the instantaneous velocity of the particle is equal to its average velocity in this interval.

Show that the following systems of linear equations has infinite number of solutions and solve:

x - y + 3z = 6,

x + 3y - 3z = -4,

5x + 3y + 3z = 10

Find the inverse of each of the following matrices $($if they exist$)$ : $\left[\begin{array}{lll}2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3\end{array}\right]$

Evaluate the following :

$\int \sqrt{\frac{9+x}{9-x}} \cdot d x$

Two cards are randomly drawn, with replacement, from a well shuffled deck of 52 playing cards. Find the probability distribution of the number of aces drawn.

Solve graphically : 5y + 3 ≤ 0

A ladder 10 metres long is leaning against a vertical wall. If the bottom of the ladder is pulled horizontally away from the wall at the rate of 1.2 metres per second, find how fast the top of the ladder is sliding down the wall, when the bottom is 6 metres away from the wall.

A bag contains 1 red and 2 green balls. One ball is drawn from the bag at random, its colour is noted, and then ball is put back in the bag. One more ball is drawn from the bag at random and its colour is also noted. Let $X$ denote the number of red balls drawn from the bag as described above. Derive the probability distribution of $X$.

Differentiate the following w. r. t. x.$\tan ^{-1}\left(\sqrt{\frac{3-x}{3+x}}\right)$