What is the integral of csc^4(8t)cot(8t) ?

The integral of csc^4(8t)cot(8t) is -1/32 csc^4(8t)+C

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integral of csc^4(8t)cot(8t)

\int csc^{4} (8 t) cot (8 t) d t

- \frac{1}{32} csc^{4} (8 t) + C

Solution steps

\int csc^{4} (8 t) cot (8 t) d t

Apply u-substitution

: \int csc^{4} (u) cot (u) \frac{1}{8} d u

= \int csc^{4} (u) cot (u) \frac{1}{8} d u

Take the constant out:

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

= \frac{1}{8} \cdot \int csc^{4} (u) cot (u) d u

Rewrite using trig identities

= \frac{1}{8} \cdot \int \frac{cos ( u )}{sin ^{5} ( u )} d u

Apply u-substitution

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

= \frac{1}{8} \cdot \int \frac{1}{v ^{5}} d v

Apply the Power Rule

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

= \frac{1}{8} (- \frac{1}{4 v ^{4}})

Substitute back

= \frac{1}{8} (- \frac{1}{4 sin ^{4} ( 8 t )})

Simplify

\frac{1}{8} (- \frac{1}{4 sin ^{4} ( 8 t )}) : - \frac{1}{32} csc^{4} (8 t)

= - \frac{1}{32} csc^{4} (8 t)

Add a constant to the solution

= - \frac{1}{32} csc^{4} (8 t) + C

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What is the integral of csc^4(8t)cot(8t) ?
The integral of csc^4(8t)cot(8t) is -1/32 csc^4(8t)+C