Evaluate the following integral:`int_0^(2pi)sin^(100)xcos^(101)x\ dx`

Explanation

Given:`int_0^(2pi)sin^(100)xcos^(101)x\ dx`

`I=int_0^(2pi)sin^(100)xcos^(101)x\ dx`

As , `int_0^(2a)f(x)dx=int_0^(a)f(x)dx+int_0^a f(2a-x)dx`

If `f(2a-x)=f(x)` then `int_0^(2a) f(x)dx=2int_0^a f(x) dx`

`I=2int_0^(pi)sin^(100)xcos^(101)x dx`

As `sin2pi-x=-sinx` and `cos2pi-x`

`=cosx` and `(-sinx)^100=sin^100x......(i)`

And `int_a^b f(x)dx=int_a^b f(a+b-x)dx`

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