How do you Integrate?

Question

#int sec^6x#

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Answer 1 · 2018-05-04T07:21:53

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

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#