Download App

Question Bank [2022] — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsQuestion Bank [2022]4 Marks
Question
$\int x \sin 2 x \cos 5 x d x$
✓

Answer

$\text { Let } I =\int x \sin 2 x \cos 5 x d x$
$=\frac{1}{2} \int x(2 \sin 2 x \cos 5 x) d x$
$=\frac{1}{2} \int x[\sin (2 x+5 x)+\sin (2 x-5 x)] d x$
$=\frac{1}{2} \int x[\sin 7 x-\sin (-3 x)] d x$
$=\frac{1}{2} \int x(\sin 7 x-\sin 3 x) d x$
$=\frac{1}{2} \int x \sin 7 x d x-\frac{1}{2} \int x \sin 3 x d x$
$=\frac{1}{2}\left[x \int \sin 7 x d x-\int\left\{\frac{ d }{ d x}(x) \int \sin 7 x d x\right\} d x\right]-\frac{1}{2}\left[x \int \sin 3 x d x-\int\left\{\frac{ d }{ d x}(x) \int \sin 3 x d x\right\} d x\right]$
$=\frac{1}{2}\left[x\left(-\frac{\cos 7 x}{7}\right)-\int 1 \cdot\left(\frac{-\cos 7 x}{7}\right) d x\right]-\frac{1}{2}\left[x\left(\frac{-\cos 3 x}{3}\right)-\int 1 \cdot\left(\frac{-\cos 3 x}{3}\right) d x\right]$
$=\frac{1}{2}\left(\frac{-x \cos 7 x}{7}+\frac{1}{7} \int \cos 7 x d x\right)-\frac{1}{2}\left(\frac{-x \cos 3 x}{3}+\frac{1}{3} \int \cos 3 x d x\right)$
$=\frac{1}{2}\left[\frac{-x \cos 7 x}{7}+\frac{1}{7}\left(\frac{\sin 7 x}{7}\right)\right]-\frac{1}{2}\left[\frac{-x \cos 3 x}{3}+\frac{1}{3}\left(\frac{\sin 3 x}{3}\right)\right]+ 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.(4 Marks)" from Question Bank [2022]→Question Bank [2022] — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Show that $\frac{d y}{d x}=\frac{y}{x}$ in the following, where a and $p$ are constants.

$x^p y^4=(x+y)^{p+4}, p \in N$

→
Find the differential equation from the relation $x^2 + 4y^2 = 4b^2$
→
Solve the following differential equations:

$xy \frac{d y}{d x}= x ^2+2 y ^2, y (1)=0$

→
Integrate the following w.r.t. x:

$\log (1+\cos x)-x \tan \left(\frac{x}{2}\right)$

→
Show that $2 \cot ^{-1} \frac{3}{2}+\sec ^{-1} \frac{13}{12}=\frac{\pi}{2}$
→
Solve the following differential equations:

$x \frac{d y}{d x}-y+x \sin \left(\frac{y}{x}\right)=0$

→
Differentiate the following w.r.t. x:

$x^e+x^x+e^x+e^e$

→
Find the area of the circle $x^2+y^2=9$, using integration.
→
Find the inverse of the matrix,
$A=\left[\begin{array}{ccc}1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1\end{array}\right]$ using elementary row transformations.
→
Find the direction cosines of the line perpendicular to the line whose direction ratios are $-2,1,-1$ and $-3,-4,1$.
→