Mathematics

# Solve $\displaystyle \int { { x }^{ 2 }.\sin {2 x }\ dx }$

$\dfrac { {- x }^{ 2 }\cos { 2x } }{ 2 } +\dfrac { 2x\sin { 2x } }{ 4 } +\dfrac { 2\cos { 2x } }{ 8 } +c$

##### SOLUTION
$\int {{x^2}.\sin 2xdx}$
$= {x^2} \times \dfrac{{ - \cos 2x}}{2} - 2x \times \dfrac{{ - \cos 2x}}{2}dx$
$= \dfrac{{ - {x^2}\cos 2x}}{2} + \int {x\cos 2xdx}$
$= \dfrac{{ - {x^2}\cos 2x}}{2} + x \times \dfrac{{\sin 2x}}{2} - \int {1 \times } \dfrac{{\sin 2x}}{2}dx$
$= \dfrac{{ - {x^2}\cos 2x}}{2} + x \times \dfrac{{\sin 2x}}{2}-\dfrac{{ 1}}{2} \times \dfrac{{ - \cos 2x}}{2} + c$
$= \dfrac{{ - {x^2}\cos 2x}}{2} + \dfrac{{x\sin 2x}}{2} + \dfrac{{\cos 2x}}{4} + c$
$= \dfrac{{ - {x^2}\cos 2x}}{2} + \dfrac{{2x\sin 2x}}{4} + \dfrac{{2\cos 2x}}{8} + c$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Assertion & Reason Hard
##### ASSERTION

$\displaystyle \int_{0}^{\pi /4}\frac{\cos x+\sin x}{\cos ^{2}x+\sin ^{4}x}dx=\frac{\pi }{4}+\frac{1}{2\sqrt{3}}\log \left ( 2+\sqrt{3} \right )=I$

##### REASON

$\displaystyle I=\int_{0}^{1}\frac{dx}{1-x^{2}+x^{4}}$

• A. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• B. Assertion is correct but Reason is incorrect
• C. Both Assertion and Reason are incorrect
• D. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Solve $\displaystyle \int{\dfrac{2x\ln \left( {{x}^{2}}-1 \right)}{\left( {{x}^{2}}-1 \right)}}dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
The value of $\displaystyle \int_{-1/2}^{1/2} \ cosx \ ln(x+\sqrt{1+x^2})dx$ is equal to ?
• A. $1$
• B. $-In2$
• C. $\sqrt{2}+In2$
• D. $0$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
The value of $\displaystyle\lim_{x\rightarrow 0}\dfrac{\displaystyle\int^{x^2}_0\cos t^2dt}{x\sin x}$ is?
• A. $1$
• B. $-1$
• C. None of these
• D. $3/2$

1 Verified Answer | Published on 17th 09, 2020

Q5 Multiple Correct Hard
The value of the integral $\displaystyle I=\int_{0}^{\pi/4} \frac{dx}{a^{2}\cos^{2}x+b^{2}\sin^{2} x}$ is
• A. $\displaystyle \frac{1}{ab}\tan^{-1}\frac{a}{b}+\frac{1}{ab}$
• B. $\displaystyle \frac{1}{ab}\tan^{-1}\frac{b}{a}(a>0,b>0)$
• C. $\displaystyle \frac{1}{ab}\tan^{-1}\frac{b}{a}(a<0,b<0)$
• D. $\displaystyle \frac{\pi}{4}(a=1,b=1)$