# Integral of 20 sec³(x) dx

#### dexstarr

##### New member
I'm having a hard time breaking this down using tri identities.

} = integral sign

20}sec^3(x) dx
20}sec^2(x) * sec(x) dx
20}[tan^2(x)-1] * sec(x) dx

after this I'm stuck. I tried letting u = tan(x) or sec(x) but i cant seem to cancel anything out.

#### SuperSonic4

##### Well-known member
MHB Math Helper
Re: Integral of 20 sec^3(x)dx

I'm having a hard time breaking this down using tri identities.

} = integral sign

20}sec^3(x) dx
20}sec^2(x) * sec(x) dx
20}[tan^2(x)-1] * sec(x) dx

after this I'm stuck. I tried letting u = tan(x) or sec(x) but i cant seem to cancel anything out.
$$\displaystyle 20 \int \frac{1}{\cos^3(x)}$$

Multiply by $$\displaystyle \dfrac{\cos(x)}{\cos(x)}$$

$$\displaystyle 20 \int \frac{\cos(x)}{\cos^4(x)}$$

$$\displaystyle 20 \int \frac{\cos(x)}{(1-\sin^2(x))^2}$$

Now use $$\displaystyle u = \sin(x)$$

#### topsquark

##### Well-known member
MHB Math Helper
Re: Integral of 20 sec^3(x)dx

Multiply by $$\displaystyle \dfrac{\cos(x)}{\cos(x)}$$
Coooool! I've never seen that method.

-Dan