What is the integral of e^xtan^3(e^x) ?

The integral of e^xtan^3(e^x) is -ln|sec(e^x)|+(sec^2(e^x))/2+C

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integral of e^xtan^3(e^x)

\int e^{x} tan^{3} (e^{x}) d x

- ln ∣ sec (e^{x}) ∣ + \frac{sec ^{2} ( e ^{x} )}{2} + C

Solution steps

\int e^{x} tan^{3} (e^{x}) d x

Apply u-substitution

: \int tan^{3} (u) d u

= \int tan^{3} (u) d u

Rewrite using trig identities

= \int (- 1 + sec^{2} (u)) tan (u) d u

Apply u-substitution

: \int \frac{- 1 + v ^{2}}{v} d v

= \int \frac{- 1 + v ^{2}}{v} d v

Expand

\frac{- 1 + v ^{2}}{v} : - \frac{1}{v} + v

= \int - \frac{1}{v} + v d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int \frac{1}{v} d v + \int v d v

\int \frac{1}{v} d v = ln ∣ v ∣

\int v d v = \frac{v ^{2}}{2}

= - ln ∣ v ∣ + \frac{v ^{2}}{2}

Substitute back

= - ln ∣ sec (e^{x}) ∣ + \frac{sec ^{2} ( e ^{x} )}{2}

Add a constant to the solution

= - ln ∣ sec (e^{x}) ∣ + \frac{sec ^{2} ( e ^{x} )}{2} + C

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What is the integral of e^xtan^3(e^x) ?
The integral of e^xtan^3(e^x) is -ln|sec(e^x)|+(sec^2(e^x))/2+C