Consider the integral
∫sec4(x)tan(x)
Now right off the bat I see two ways of solving this.
- Let u=sec(x)
2.Use integration by parts
Now doing the first way results in the integrand looking like
∫u3du=14sec4(x)+C
Which is correct but it's not the answer I'm looking for, so instead we'll do it the second way.
∫sec2(x)⋅sec2(x)tan(x)dx
∫(tan2(x)+1)sec2(x)tan(x)dx
∫sec2(x)tan3(x)+sec2(x)tan(x)dx
Now this is where I got stuck, because I don't know whether to continue with Pythagorean identities or to factor a term out and solve for that. Or perhaps even break the two up and create two integrals.
Answer
Write your integrand in the formtan(x)(tan2(x)+1)sec2(x) and substitute u=tan(x) and you will get ∫u(u2+1)du
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