Mathematics

# Let ${I_n} = \int_0^\pi {\dfrac{{{{\sin }^2}\left( {nx} \right)}}{{{{\sin }^2}x}}} \,dx\,\,,\,n \in N$, then

${I_{n + 2}} + {I_n} = {21_{n + 1}}$

##### SOLUTION
${ I }_{ n }=\displaystyle \int _{ 0 }^{ \pi }{ \frac { \sin ^{ 2 }{ nx } }{ \sin ^{ 2 }{ x } } dx } ;\quad { I }_{ n+1 }=\displaystyle \int _{ 0 }^{ \pi }{ \frac { \sin ^{ 2 }{ \left( n+1 \right) } }{ \sin ^{ 2 }{ x } } dx }$

${ I }_{ n+1 }-{ I }_{ n }=\displaystyle \int _{ 0 }^{ \pi }{ \frac { \sin ^{ 2 }{ \left( n+1 \right) x- } \sin ^{ 2 }{ nx } }{ \sin ^{ 2 }{ x } } }$

$=\displaystyle\int _{ 0 }^{ \pi }{ \frac { \sin ^{ 2 }{ \left( 2n+1 \right) x. } \sin { x } }{ \sin ^{ 2 }{ x } } } dx$

$=\displaystyle \int _{ 0 }^{ \pi }{ \frac { \sin ^{ 2 }{ \left( 2n+1 \right) x } }{ \sin { x } } } dx$

$I_{n+2}-I_{n+1}=\displaystyle \int _{ 0 }^{ \pi }{ \frac { \sin { \left( 2n+3 \right) x } }{ \sin { x } } } dx$

$I_{n+2}-2I_{n+1}+I_n=\displaystyle \int _{ 0 }^{ \pi }{ \frac { \sin { \left( 2n+3 \right) x- } \sin { \left( 2n+1 \right) x } }{ \sin { x } } } dx$

$=\displaystyle\int _{ 0 }^{ \pi }{ \frac { \cos { \left( 2n+3 \right) x- } \sin { x } }{ \sin { x } } } dx$

$=\displaystyle\int _{ 0 }^{ \pi }{ 2\cos 2(n+1)x dx}$

$=\left. 2\dfrac{\sin (2n+2)x}{2x+2}\right|_{0}^{\pi}=0$

so we have
$I_{n+2}+I_n=2I_{n+1}$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Subjective Hard
Solve $\displaystyle\int { \dfrac { { x }^{ 2 }+1 }{ { x }^{ 2 }-5x+6 } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \underset{0}{\overset{\pi}{\int}} \dfrac{x \, dx}{ 1 + \sin \, x}$ =
• A. $\dfrac{\pi}{6}$
• B. $\dfrac{\pi}{3}$
• C. Cannot be valued
• D. $\pi$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
The value of $\displaystyle \int \sqrt{2}\left ( \frac{\sin x}{\sin \left ( x-\dfrac {\pi }4 \right )} \right )dx$ is
• A. $\displaystyle x-\log \left | \sin \left ( x-\frac{\pi }{4} \right ) \right |+c$
• B. $\displaystyle x+\log \left | \cos \left ( x-\frac{\pi }{4} \right ) \right |+c$
• C. $\displaystyle x-\log \left | \cos \left ( x-\frac{\pi }{4} \right ) \right |+c$
• D. $\displaystyle x+\log \left | \sin \left ( x-\frac{\pi }{4} \right ) \right |+c$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Solve $\displaystyle \int {\tan x\,\ln \left( {\cos x} \right)} \,dx$
• A. $\dfrac {\ln^2(\cos x)}{2}+C$
• B. $\dfrac {\ln^2(\sin x)}{2}+C$
• C. $\dfrac {\ln(\cos x)}{2}+C$
• D. $-\dfrac {(\ln(\cos x))^2}{2}+C$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
$\displaystyle \int_2^3\dfrac{dx}{x^2-1}$
• A. $log\dfrac{3}{2}$
• B. $2log\dfrac{3}{2}$
• C. $\log\dfrac{3}{2}$
• D. $\dfrac{1}{2}log\dfrac{3}{2}$