Evaluate the following integrals: 2x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\text{x}\sin2\text{x dx}$

✓

Answer

Let $\text{I}=\int\text{x}\sin2\text{x dx}$
Using integration by parts,
$\text{I}=\text{x}\int\sin2\text{x dx}-\int(1\int\sin2\text{x dx})\text{dx}$
$=\text{x}\Big(-\frac{\cos2\text{x}}{2}\Big)-\int\Big(-\frac{\cos2\text{x}}{2}\Big)\text{dx}$
$=-\frac{\text{x}}{2}\cos2\text{x}+\frac{1}{2}\int\cos2\text{x dx}$
$=-\frac{\text{x}}{2}\cos2\text{x}+\frac{1}{2}\frac{\sin2\text{x}}{2}+\text{C}$
$\text{I}=-\frac{\text{x}}{2}\cos2\text{x}+\frac{1}{4}\sin2\text{x}+\text{C}$

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 "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

Solve the following system of homogeneous linear equations:

2x + 3y + 4z = 0,

2x - y + 3z = 0

Write the position vector of the point which divdes the join of the points with position vectors $3\vec{\text{a}} - 2\vec{\text{b}} $ and $2\vec{\text{a}} + 3\vec{\text{b}} $ in the ratio 2 : 1.

Evaluate the following definite integrals:$\int_{0}^\limits{\frac{\pi}{4}}\text{x}^2\sin\text{x}\text{ dx}$

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

Integrate the following w.r.t. x:

$\sec ^2 x \sqrt{7+2 \tan x-\tan ^2 x}$

Find the values of x, for which the function $f(x) = x^3 + 12x^2 + 36? + 6$ is monotonically decreasing

In the following, find the value of the constant k so that the given function is continuous at the indicated point:
$\text{f(x)}=\begin{cases}\frac{1-\cos2\text{kx}}{\text{x}^2},&\text{if}\text{ x}\neq0\\8,&\text{if}\text{ x}=0\end{cases}\text{at x}=0$

Find $\frac{d y}{d x}$ if : $x=\sqrt{1-t^2}, y=\sin ^{-1} t$

If the angles $A, B, C$ of $\triangle ABC$ are in A.P. and its sides $a, b, c$ are in G.P., then show that $a^2, b^2, c^2$ are in A.P.

Integrate the following functions w.r.t x:

$\frac{\sin x \cos ^3 x}{1+\cos ^2 x}$