Finding the value of integral.

[Raghav Gupta]

Homework Statement

$$ \int_0^{\pi} \frac{x(sinx)^{2n}}{(sinx)^{2n}+(cosx)^{2n}} $$ =
A) π²
B) 2π²
C) π²/4
D) π/2

Homework Equations

$$ \int_0^π f(x)dx = \int_0^π f(a-x)dx $$

The Attempt at a Solution

If we put n = 1, we get the C option π²/4
But is it true for all n?

 

[certainly]

Remember that $\int_0^{2a} f(x)dx=2\int_0^a f(x)dx$ iff $f(2a-x)=f(x)$ and then consider the integral $\int_0^{\pi} \frac{(sinx)^{2n}}{(sinx)^{2n}+(cosx)^{2n}}$ using the rule you mentioned.

 

[Raghav Gupta]

Remember that $\int_0^{2a} f(x)dx=2\int_0^a f(x)dx$ iff $f(2a-x)=f(x)$ and then consider the integral $\int_0^{\pi} \frac{(sinx)^{2n}}{(sinx)^{2n}+(cosx)^{2n}}$ using the rule you mentioned.

Thanks, I got it certainly Calculus Cuthbert.

 

[certainly]

muffins from the cupboard ? no, no dear fellow I'm not hungry.