What's the integral of $int {[(secx)^3] tanx} dx$ ?

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Answer 1 · 2016-10-19T23:44:36

$sec^3x/3+C$

$I=int(sec^3x)(tanx)dx$

We should be aiming to find one of two things:

An integrand composed of just $secx$ functions with one $secxtanx$ , the derivative of $secx$ .
An integrand composed of just $tanx$ functions with one $sec^2x$ , the derivative of $tanx$ .

Here, we see we can easily peel of one $secx$ from $sec^3x$ to combine it into $secxtanx$ , leaving just other $secx$ functions.

Rewriting the function:

$I=int(sec^2x)(secxtanx)dx$

We can now use substitution. Let $u=secx$ so that $du=(secxtanx)dx$ .

Thus, the integral becomes:

$I=intu^2du$

Which, through the power rule for integration, becomes:

$I=u^3/3+C$

$I=sec^3x/3+C$

What's the integral of int {[(secx)^3] tanx} dxint {[(secx)^3] tanx} dx?