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How do you integrate #int tan^3xsec^2x# using substitution?

Answer 1

Emilia Aguilar

Use the substitution #u=tanx#.

Let

#I=inttan^3xsec^2dx#

Apply the substitution #u=tanx#:

#I=intu^3du#

Integrate directly:

#I=1/4u^4+C#

Reverse the substitution:

#I=1/4tan^4x+C#

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Answer 2

Christopher Allers

To integrate ∫tan^3(x)sec^2(x) dx using substitution:

Let u = tan(x), then du = sec^2(x) dx.
Rewrite the integral in terms of u: ∫u^3 du.
Integrate the new expression: (1/4)u^4 + C.
Replace u with tan(x): (1/4)tan^4(x) + C.

So, the integral of tan^3(x)sec^2(x) dx is (1/4)tan^4(x) + C.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.