Integrate with sin cos #3

Calculus Level 5

If $\large{\displaystyle \int^{\frac{\pi}{2}}_{0} f(x) dx=\frac{A}{B}\pi}$

where $A,B$ are co prime natural numbers.

Find $A+B$

given that

$\large{f\left( x \right) =5\int _{ 0 }^{ x }{ \frac { 6 }{ \sin ^{ 2 }{ \left( x \right) } } \int _{ 0 }^{ x }{ .......... } \int _{ 0 }^{ x }{ \frac { 2014 }{ \sin ^{ 2 }{ \left( x \right) } } \int _{ 0 }^{ x }{ \frac { 2015 }{ \sin ^{ 2 }{ \left( x \right) } } \int _{ 0 }^{ x }{ \frac { 2016 }{ \sin ^{ 2 }{ \left( x \right) } } } } \int _{ 0 }^{ x }{ \sin { \left( 2017x \right) \sin ^{ 2015 }{ \left( x \right) } { \left( dx \right) }^{ 2012 } } } } } }$

Try my set

The answer is 65.

1 solution

Tanishq Varshney
Sep 13, 2015

$\large{\int \sin ((n+2)x) \sin^{n} (x) dx}$

$\large{\int \sin((n+1)x+x) \sin^{n} (x) dx}$

$\large{\frac{1}{n+1} \int \left ((n+1)\sin ((n+1)x) \cos (x) \sin^{n} (x)+(n+1)\cos((n+1)x) \sin^{n+1} (x) \right )dx}$

$\large{\frac{1}{n+1} \int d(\sin((n+1)x) \sin^{n+1} (x))}$

$\large{=\frac{\sin((n+1)x) \sin^{n+1} (x)}{n+1}+c}$

on applying the limits we get

$\large{=\frac{\sin((n+1)x) \sin^{n+1} (x)}{n+1}}$

we observe that after multiplying $\frac{n+1}{\sin^{2} (x)}$

we begin again from the very first step

in the end we get

$\large{f(x)=\sin (5x) \sin^{5} (x) }$

How do we integrate this??

Vidit Kulshreshtha - 4 years, 2 months ago

Integrate with sin cos #3

Try my set

The answer is 65.

1 solution

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