How do you Integrate?

int sec^6x

1 Answer
May 4, 2018

I=1/5tan^5(x)+2/3tan^3(x)+tan(x)+C

Explanation:

We want to integrate

I=intsec^6(x)dx

Rewrite the integrand using the trig identity

color(blue)(sec^2(x)=1+tan^2(x)

I=intsec^2(x)(sec^2(x))^2dx

color(white)(I)=intsec^2(x)(1+tan^2(x))^2dx

Make a substitution color(blue)(u=tan(x)=>du=sec^2(x)dx

I=int(1+u^2)^2du

color(white)(I)=intu^4+2u^2+1du

color(white)(I)=1/5u^5+2/3u^3+u+C

Substitute back color(blue)(u=tan(x)

I=1/5tan^5(x)+2/3tan^3(x)+tan(x)+C