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

The integral of 4csc^4(x)cot(x) is -csc^4(x)+C

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integral of 4csc^4(x)cot(x)

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

- csc^{4} (x) + C

Solution steps

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

Take the constant out:

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

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

Rewrite using trig identities

= 4 \cdot \int \frac{cos ( x )}{sin ^{5} ( x )} d x

Apply u-substitution

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

= 4 \cdot \int \frac{1}{u ^{5}} d u

Apply the Power Rule

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

= 4 (- \frac{1}{4 u ^{4}})

Substitute back

u = sin (x)

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

Simplify

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

= - csc^{4} (x)

Add a constant to the solution

= - csc^{4} (x) + C

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What is the integral of 4csc^4(x)cot(x) ?
The integral of 4csc^4(x)cot(x) is -csc^4(x)+C