What is the integral of cos(x)csc^8(x) ?

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

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integral of cos(x)csc^8(x)

\int cos (x) csc^{8} (x) d x

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

Solution steps

\int cos (x) csc^{8} (x) d x

Rewrite using trig identities

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

Apply u-substitution

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

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

Apply the Power Rule

: - \frac{1}{7 u ^{7}}

= - \frac{1}{7 u ^{7}}

Substitute back

u = sin (x)

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

Simplify

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

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

Add a constant to the solution

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

\frac{d}{d x} (csc^{5} (x))

\int sin (7 x) cos (3 x) d x

\frac{d y}{d x} = 0.5 (y - 200)

\frac{d}{d x} (tanh (x))

x \to 2 lim (\frac{∣ x - 2 ∣}{x})

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