What is the integral of (sin(x))/(cos^6(x)) ?

The integral of (sin(x))/(cos^6(x)) is (sec^5(x))/5+C

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integral of (sin(x))/(cos^6(x))

\int \frac{sin ( x )}{cos ^{6} ( x )} d x

\frac{sec ^{5} ( x )}{5} + C

Solution steps

\int \frac{sin ( x )}{cos ^{6} ( x )} d x

Rewrite using trig identities

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

Use the basic trigonometric identity:

\frac{1}{cos ( x )} = sec (x)

= \int sec^{5} (x) tan (x) d x

Apply u-substitution

: \int u^{4} d u

= \int u^{4} d u

Apply the Power Rule

: \frac{u ^{5}}{5}

= \frac{u ^{5}}{5}

Substitute back

u = sec (x)

= \frac{sec ^{5} ( x )}{5}

Add a constant to the solution

= \frac{sec ^{5} ( x )}{5} + C

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