What is the integral of (cos^3(x))/(sin^8(x)) ?

The integral of (cos^3(x))/(sin^8(x)) is -1/7 csc^7(x)+1/5 csc^5(x)+C

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integral of (cos^3(x))/(sin^8(x))

\int \frac{cos ^{3} ( x )}{sin ^{8} ( x )} d x

- \frac{1}{7} csc^{7} (x) + \frac{1}{5} csc^{5} (x) + C

Solution steps

\int \frac{cos ^{3} ( x )}{sin ^{8} ( x )} d x

Rewrite using trig identities

= \int \frac{( 1 - sin ^{2} ( x ) ) cos ( x )}{sin ^{8} ( x )} d x

Apply u-substitution

: \int \frac{1 - u ^{2}}{u ^{8}} d u

= \int \frac{1 - u ^{2}}{u ^{8}} d u

Expand

\frac{1 - u ^{2}}{u ^{8}} : \frac{1}{u ^{8}} - \frac{1}{u ^{6}}

= \int \frac{1}{u ^{8}} - \frac{1}{u ^{6}} d u

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}{u ^{8}} d u - \int \frac{1}{u ^{6}} d u

\int \frac{1}{u ^{8}} d u = - \frac{1}{7 u ^{7}}

\int \frac{1}{u ^{6}} d u = - \frac{1}{5 u ^{5}}

= - \frac{1}{7 u ^{7}} - (- \frac{1}{5 u ^{5}})

Substitute back

u = sin (x)

= - \frac{1}{7 sin ^{7} ( x )} - (- \frac{1}{5 sin ^{5} ( x )})

Simplify

- \frac{1}{7 sin ^{7} ( x )} - (- \frac{1}{5 sin ^{5} ( x )}) : - \frac{1}{7} csc^{7} (x) + \frac{1}{5} csc^{5} (x)

= - \frac{1}{7} csc^{7} (x) + \frac{1}{5} csc^{5} (x)

Add a constant to the solution

= - \frac{1}{7} csc^{7} (x) + \frac{1}{5} csc^{5} (x) + C

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What is the integral of (cos^3(x))/(sin^8(x)) ?
The integral of (cos^3(x))/(sin^8(x)) is -1/7 csc^7(x)+1/5 csc^5(x)+C