What is the integral of (sec(θ)tan^5(θ))/(cos(θ)) ?

The integral of (sec(θ)tan^5(θ))/(cos(θ)) is (tan^6(θ))/6+C

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integral of (sec(θ)tan^5(θ))/(cos(θ))

\int \frac{sec ( θ ) tan ^{5} ( θ )}{cos ( θ )} d θ

\frac{tan ^{6} ( θ )}{6} + C

Solution steps

\int \frac{sec ( θ ) tan ^{5} ( θ )}{cos ( θ )} d θ

Rewrite using trig identities

= \int \frac{tan ^{5} ( θ )}{cos ^{2} ( θ )} d θ

Rewrite using trig identities

= \int tan^{5} (θ) sec^{2} (θ) d θ

Rewrite using trig identities

= \int (1 + tan^{2} (θ)) tan^{5} (θ) d θ

Apply u-substitution

: \int u^{5} d u

= \int u^{5} d u

Apply the Power Rule

: \frac{u ^{6}}{6}

= \frac{u ^{6}}{6}

Substitute back

u = tan (θ)

= \frac{tan ^{6} ( θ )}{6}

Add a constant to the solution

= \frac{tan ^{6} ( θ )}{6} + C

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What is the integral of (sec(θ)tan^5(θ))/(cos(θ)) ?
The integral of (sec(θ)tan^5(θ))/(cos(θ)) is (tan^6(θ))/6+C