Integrating Tan Squared Sec Squared: Solving the Challenging Integral

[Mentallic]

Homework Statement

Find \int{tan^2xsec^2xdx}

Homework Equations

tan^2x=sec^2x-1 (1)

The Attempt at a Solution

Using (1): \int{(sec^4x-sec^2x)dx}

Now, \int{sec^4xdx}-\int{sec^2xdx}=\int{sec^4xdx}-tanx+c

I can't figure out how to solve \int{sec^4xdx} though.

 

[Dick]

The derivative of tan(x) is sec^2(x). Mentallic, this is a simple substitution. u=tan(x).

 

[Mentallic]

Oh yeah... ugh I feel like such an idiot.

u=tanx

\int{sec^4xdx}=\int{(1+u^2)du}

Thanks Dick.

 

[Dick]

Oh yeah... ugh I feel like such an idiot.

u=tanx

\int{sec^4xdx}=\int{(1+u^2)du}

Thanks Dick.

Sure, but why don't you use that substitution in the original integral?

 

[Mentallic]

Oh, you mean like \int{u^2du} ? Yeah because I love to make things harder for myself