Download App

Indefinite Integrals — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals5 Marks
Question
Evaluate the following integrals:
$\int\text{x}\sin\text{x}\cos2\text{x dx}$
✓

Answer

$\int\text{x}.\cos2\text{x}\sin\text{x dx}$
$=\frac{1}{2}\int\text{x}(2\cos2\text{x}\sin\text{x})\text{dx}$ $\big[\therefore2\cos\text{A}\sin\text{B}=\sin(\text{A+B})-\sin(\text{A}-\text{B})\big]$
$=\frac{1}{2}\int\text{x}(\sin3\text{x}-\sin\text{x})\text{dx}$
$=\frac{1}{2}\int\text{x}\sin3\text{x dx}-\frac{1}{2}\int\text{x}\sin\text{x dx}$
$=\frac{1}{2}\int\text{x}\sin3\text{x dx}-\frac{1}{2}\int\text{x}\sin\text{x dx}$
$=\frac{1}{2}\Big[\text{x}\int\sin3\text{x dx}-\int\Big\{\frac{\text{x}}{\text{dx}}(\text{x})\int\sin3\text{x dx}\Big\}\text{dx}\Big]\\-\frac{1}{2}\Big[\text{x}\int\sin\text{x dx}-\int\Big\{\frac{\text{d}}{\text{dx}}(\text{x})\int\sin\text{x dx}\Big\}\text{dx}\Big]$
$=\frac{1}{2}\Big[\text{x}\Big(\frac{-\cos3\text{x}}{3}\Big)-\int1\Big(\frac{-\cos3\text{x}}{3}\Big)\text{dx}\Big]\\-\frac{1}{2}\big[\text{x}(-\cos\text{x})-\int1(-\cos\text{x})\text{dx}\big]$
$=\frac{1}{2}\Big[\text{x}\Big(\frac{-\cos3\text{x}}{3}\Big)+\frac{1}{9}\sin3\text{x}\big]-\frac{1}{2}\big[\text{x}(-\cos\text{x})+\sin\text{x}\big]$
$=-\frac{\text{x}\cos3\text{x}}{6}+\frac{\sin3\text{x}}{18}+\frac{\text{x}\cos\text{x}}{2}-\frac{\sin\text{x}}{2}+\text{C}$

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.(5 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

Evaluate:
$\begin{vmatrix}\text{a}&\text{b}+\text{c}&\text{a}^2\\\text{b}&\text{c}+\text{a}&\text{b}^2\\\text{c}&\text{a}+\text{b}&\text{c}^2\end{vmatrix}$
→
Find the vector equation of the plane which is at a distance of $\frac{6}{\sqrt{29}}$ from the origin and its normal vector from the origin is $2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}}$ Also, find its cartesian form.
→
Evaluate the following integrals:
$\int\frac{\cos^5\text{x}}{\sin\text{x}}\text{ dx}$
→
Solve the following differential equations:$\frac{\text{dy}}{\text{dx}}=\frac{\text{x}(2\log\text{x}+1)}{\sin\text{y + y}\cos\text{y}}$
→
Evaluate the following intregals: $\int\frac{1}{\sqrt{3}\sin\text{x}+\cos\text{x}}\ \text{dx}$
→
Differentiate $\sin^{-1}\Big(2\text{ax}\sqrt{1-\text{a}^2\text{x}^2}\Big)$ with respect to $\sqrt{1-\text{a}^2\text{x}^2},$ if $-\frac{1}{\sqrt{2}}<\text{ax}<\frac{1}{\sqrt{2}}$.
→
Evaluate the following integrals:
$\int\text{x}^2\text{e}^{\text{x}^3}\cos\text{x}^3\text{dx}$
→
Examine the continuity of the function
$\text{f}\text{(x)}=\begin{cases}3\text{x}-2 &, \text{ if x} \leq 0\\\text{x}+1 &, \text{ if x} > 0\end{cases}\text{at x}=0$
Also sketch the graph of this function.
→
Evaluate the following integrals as limit of sum:$\int\limits^2_{0}\big(\text{x}^2+1\big)\text{dx}$
→
$\vec{\text{a}},\vec{\text{b}}$ and $\vec{\text{c}}$ are the position vectors of points A, B and C respectively, prove that:
$\vec{\text{a}}\times\vec{\text{b}}+\vec{\text{b}}\times\vec{\text{c}}+\vec{\text{c}}\times\vec{\text{a}}$ is a vector perpendicular to the plane of triangle ABC.
→