Question

Question #c78e9

Answer 1

`int_0^(pi/4) (d theta)/(1+sin2theta) = 1/2`

Explanation:

Using parametric formulas:

`sin2theta = (2tantheta)/(1+tan^2theta)`

`int_0^(pi/4) (d theta)/(1+sin2theta) = int_0^(pi/4) (d theta)/(1+(2tantheta)/(1+tan^2theta))`

`int_0^(pi/4) (d theta)/(1+sin2theta) = int_0^(pi/4) ((1+tan^2theta)d theta)/(2tantheta+1+tan^2theta) = int_0^(pi/4) ((1+tan^2theta)d theta)/(1+tantheta)^2`

As:

`1+tan^2theta = sec^2theta`

and

`d/(d theta) (1+tan theta) = sec^2 theta`

we have:

`int_0^(pi/4) (d theta)/(1+sin2theta) = int_0^(pi/4) (sec^2theta d theta)/(1+tan theta)^2 = int_0^(pi/4) (d (1+tan theta))/(1+tan theta)^2`

`int_0^(pi/4) (d theta)/(1+sin2theta) = [-1/(1+tan theta)]_0^(pi/4) =-1/2+1=1/2`