Question

Integral of the power products following integral: `int_0^(pi/2)sen^2(theta/3)d theta` ?

`int_0^(pi/2)sen^2(theta/3)d theta`

Answer 1

`pi/4-(3sqrt3)/8`

Explanation:

Recall the identity

`sin^2theta=1/2(1-cos2theta)`

Then,

`sin^2(theta/3)=1/2(1-cos((2theta)/3)`

Thus, we have

`1/2int_0^(pi/2)(1-cos((2theta)/3))d theta`

Let's determine the general antiderivative.

`int1-cos((2theta)/3)d theta=theta-3/2sin((2theta)/3)`, as

`intcosathetad theta=1/asinatheta`

So,

`1/2int_0^(pi/2)(1-cos((2theta)/3))d theta=1/2(theta-3/2sin((2theta)/3)|_0^(pi/2)`

Evaluate:

`1/2(pi/2-3/2sin((2pi)/6)-0+3/2sin(0))=1/2(pi/2-3/2sin(pi/3))=1/2(pi/2-3/2(sqrt3/2))=1/2(pi/2-(3sqrt3)/4)=pi/4-(3sqrt3)/8`