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