Evaluate the following integrals: ^2 dx - Vidyadip

Question

Evaluate the following integrals:$\int\text{x}^2\cos\text{x dx}$

✓

Answer

Let $\text{I}=\int\text{x}^2\cos\text{x dx}$
Using integration by parts,
$\text{I}=\text{x}^2\int\cos\text{x dx}-\int(2\text{x}\int\cos\text{x dx})\text{dx}$
$=\text{x}^2\sin\text{x}-2\int\text{x}\sin\text{x dx}$
$=\text{x}^2\sin\text{x}-2[\text{x}\int\sin\text{x dx}-\int(1\int\sin\text{x dx})\text{dx}]$
$=\text{x}^2\sin\text{x}-2[\text{x}(-\cos\text{x})-\int(-\cos\text{x})\text{dx}]$
$=\text{x}^2\sin\text{x}+2\text{x}\cos\text{x}-2\int(\cos\text{x})\text{dx}$
$\text{I}=\text{x}^2\sin\text{x}+2\text{x}\cos\text{x}-2\sin\text{x}+\text{C}$

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 Question" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Show that the relation $R$ in the set $A$ of points in a plane given by $R = \{(P, Q) :$ distance of the point $P$ from the origin is same as the distance of the point $Q$ from the origin$\}$, is an equivalence relation. Further, show that the set of all points related to a point $P \ne (0, 0)$ is the circle passing through $P$ with origin as centre.

Evaluate the following integrals:
$\int\limits^{\text{a}}_0\frac{\sqrt{\text{x}}}{\sqrt{\text{x}}+\sqrt{\text{a}-\text{x}}}\text{ dx}$

If $\text{y}=\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big),$ write the value of $\frac{\text{dy}}{\text{dx}}\text{ for x}>1.$

$\int\frac{1}{\sqrt{\text{x}+1}+\sqrt{\text{x}}}\text{dx}$

Four cards are drawn simultaneously from a well shuffled pack of 52 playing cards. Find the probability distribution of the number of aces.

Find the matrix A such that
$ \begin{bmatrix}4\\1\\3\end{bmatrix}\text{A}=\begin{bmatrix}-4&8&4\\-1&2&1\\-3&6&3\end{bmatrix}$

Evaluate the following integrals:$\int\frac{\text{x}+5}{3\text{x}^2+13\text{x}-10}\text{ dx}$

Find the binomial distribution for which the mean is 4 and variance 3.

If $\log\text{y}=\tan^{-1}$ show that $(1+\text{x}^2)\text{y}_2+(2\text{x}-1)\text{y}_1=0$

Sketch the graph of y = |x + 3| and evaluate $\int_{ - 6}^0 {\left| {x + 3} \right|dx} $