What is the integral of (cos(ax))/(sin^5(ax)) ?

The integral of (cos(ax))/(sin^5(ax)) is -1/(4asin^4(ax))+C

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integral of (cos(ax))/(sin^5(ax))

\int \frac{cos ( a x )}{sin ^{5} ( a x )} d x

- \frac{1}{4 a sin ^{4} ( a x )} + C

Solution steps

\int \frac{cos ( a x )}{sin ^{5} ( a x )} d x

Rewrite using trig identities

= \int \frac{cot ( a x )}{sin ^{4} ( a x )} d x

Use the basic trigonometric identity:

\frac{1}{sin ( x )} = csc (x)

= \int csc^{4} (a x) cot (a x) d x

Apply u-substitution

: \int \frac{csc ^{4} ( u ) cot ( u )}{a} d u

= \int \frac{csc ^{4} ( u ) cot ( u )}{a} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{a} \cdot \int csc^{4} (u) cot (u) d u

Rewrite using trig identities

= \frac{1}{a} \cdot \int \frac{cos ( u )}{sin ^{5} ( u )} d u

Apply u-substitution

: \int \frac{1}{v ^{5}} d v

= \frac{1}{a} \cdot \int \frac{1}{v ^{5}} d v

Apply the Power Rule

: - \frac{1}{4 v ^{4}}

= \frac{1}{a} (- \frac{1}{4 v ^{4}})

Substitute back

= \frac{1}{a} (- \frac{1}{4 sin ^{4} ( a x )})

Simplify

\frac{1}{a} (- \frac{1}{4 sin ^{4} ( a x )}) : - \frac{1}{4 a sin ^{4} ( a x )}

= - \frac{1}{4 a sin ^{4} ( a x )}

Add a constant to the solution

= - \frac{1}{4 a sin ^{4} ( a x )} + C

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What is the integral of (cos(ax))/(sin^5(ax)) ?
The integral of (cos(ax))/(sin^5(ax)) is -1/(4asin^4(ax))+C