What is the integral of (cos^5(17x))/(sin^3(17x)) ?

The integral of (cos^5(17x))/(sin^3(17x)) is 1/34 (-csc^2(17x)-4ln|sin(17x)|+sin^2(17x))+C

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integral of (cos^5(17x))/(sin^3(17x))

\int \frac{cos ^{5} ( 17 x )}{sin ^{3} ( 17 x )} d x

\frac{1}{34} (- csc^{2} (17 x) - 4 ln ∣ sin (17 x) ∣ + sin^{2} (17 x)) + C

Solution steps

\int \frac{cos ^{5} ( 17 x )}{sin ^{3} ( 17 x )} d x

Apply u-substitution

: \int \frac{cos ^{5} ( u )}{17 sin ^{3} ( u )} d u

= \int \frac{cos ^{5} ( u )}{17 sin ^{3} ( u )} d u

Take the constant out:

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

= \frac{1}{17} \cdot \int \frac{cos ^{5} ( u )}{sin ^{3} ( u )} d u

Rewrite using trig identities

= \frac{1}{17} \cdot \int \frac{( 1 - sin ^{2} ( u ) ) ^{2} cos ( u )}{sin ^{3} ( u )} d u

Apply u-substitution

: \int \frac{( 1 - v ^{2} ) ^{2}}{v ^{3}} d v

= \frac{1}{17} \cdot \int \frac{( 1 - v ^{2} ) ^{2}}{v ^{3}} d v

Expand

\frac{( 1 - v ^{2} ) ^{2}}{v ^{3}} : \frac{1}{v ^{3}} - \frac{2}{v} + v

= \frac{1}{17} \cdot \int \frac{1}{v ^{3}} - \frac{2}{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

= \frac{1}{17} (\int \frac{1}{v ^{3}} d v - \int \frac{2}{v} d v + \int v d v)

\int \frac{1}{v ^{3}} d v = - \frac{1}{2 v ^{2}}

\int \frac{2}{v} d v = 2 ln ∣ v ∣

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

= \frac{1}{17} (- \frac{1}{2 v ^{2}} - 2 ln ∣ v ∣ + \frac{v ^{2}}{2})

Substitute back

= \frac{1}{17} (- \frac{1}{2 sin ^{2} ( 17 x )} - 2 ln ∣ sin (17 x) ∣ + \frac{sin ^{2} ( 17 x )}{2})

Simplify

\frac{1}{17} (- \frac{1}{2 sin ^{2} ( 17 x )} - 2 ln ∣ sin (17 x) ∣ + \frac{sin ^{2} ( 17 x )}{2}) : \frac{1}{34} (- csc^{2} (17 x) - 4 ln ∣ sin (17 x) ∣ + sin^{2} (17 x))

= \frac{1}{34} (- csc^{2} (17 x) - 4 ln ∣ sin (17 x) ∣ + sin^{2} (17 x))

Add a constant to the solution

= \frac{1}{34} (- csc^{2} (17 x) - 4 ln ∣ sin (17 x) ∣ + sin^{2} (17 x)) + C

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What is the integral of (cos^5(17x))/(sin^3(17x)) ?
The integral of (cos^5(17x))/(sin^3(17x)) is 1/34 (-csc^2(17x)-4ln|sin(17x)|+sin^2(17x))+C