`int_0^(pi/4) sec^4(theta) tan^4(theta) d theta` Evaluate the integral

`int_0^(pi/4)sec^4(theta)tan^4(theta)d theta`

Let's evaluate the indefinite integral by rewriting the integrand as,

`intsec^3(theta)tan^4(theta)d theta=intsec^2(theta)sec^2(theta)tan^4(theta)d theta`

Now use the identity:`1+tan^2(x)=sec^2(x)`

`=int(1+tan^2(theta))sec^2(theta)tan^4(theta)d theta`

Now apply integral substitution,

Let `u=tan(theta)`

`=>du=sec^2(theta)d theta`

`=int(1+u^2)u^4du`

`=int(u^4+u^6)du`

`=intu^4du+intu^6du`

`=u^5/5+u^7/7`

Substitute back `u=tan(theta)`

`=1/5tan^5(theta)+1/7tan^7(theta)`

Add a constant to the solution,

`=1/5tan^5(theta)+1/7tan^7(theta)+C`

Now let's evaluate the definite integral,

`int_0^(pi/4)sec^4(theta)tan^4(theta)d theta=[1/5tan^5(theta)+1/7tan^7(theta)]_0^(pi/4)`

`=[1/5tan^5(pi/4)+1/7tan^7(pi/4)]-[1/5tan^5(0)+1/7tan^7(0)]`

`=[1/5+1/7]-[0]`

`=[(7+5)/35]`

`=12/35`