Need help in deriving this reduction formula

[hms.tech]

It might be difficult for you to read this integral in non latex form, but i'll try my best.
As i don't know how to write this in latex form, assume "for this problem" that I(n) is pronounced as "I subscript n" or nth term of I.

Homework Statement

I(n)=∫ (sinx)^n dx [with limits of the integral as : from zero (0) to ∏/2 (pi/2)

Using the above equation, it is required to prove that :

I(n+2)= I(n) * (n+1)/(n+2) [again , I(n) means I subscript n ie nth term of a sequence]

Homework Equations

the formula for integration by parts

The Attempt at a Solution

I have tried to integrate it by parts using various ways but all of them failed to prove the required result.
One of them was :
∫ [sin^-2(x)*(sin(x))^(n+2)] dx [with the same limits ofcourse]

even after subsituting 1-cos^2(x) for sin^2(x) the problem could not be solved,
please help me solve it.

 

[gomunkul51]

I tried to understand but it was too late,
so please:

http://www.codecogs.com/latex/eqneditor.php

 

[SammyS]

It might be difficult for you to read this integral in non latex form, but I'll try my best.
As i don't know how to write this in latex form, assume "for this problem" that I(n) is pronounced as "I subscript n" or nth term of I.

Homework Statement

I(n)=∫ (sinx)^n dx [with limits of the integral as : from zero (0) to ∏/2 (pi/2)

Using the above equation, it is required to prove that :

I(n+2)= I(n) * (n+1)/(n+2) [again , I(n) means I subscript n ie nth term of a sequence]

Homework Equations

the formula for integration by parts

The Attempt at a Solution

I have tried to integrate it by parts using various ways but all of them failed to prove the required result.
One of them was :
∫ [sin^-2(x)*(sin(x))^(n+2)] dx [with the same limits of course]

even after substituting 1-cos^2(x) for sin^2(x) the problem could not be solved,
please help me solve it.

Start with I_n+2.

Even without LaTeX, you can do _SUBSCRIPTS and ^SUPERSCRIPTS by using the X₂ and X² buttons in the 'Go Advanced' message window.​

\displaystyle I_{n+2}=\int\sin^{n+2}(x)\,dx=\int(1-\cos^2(x))\sin^{n}(x)\,dx=I_n-\int\cos^2(x)\sin^{n}(x)\,dx

Evaluate that last integral using integration by parts.

u=cos(x), dv=cos(x)sinⁿ(x) dx​