What is the integral of csc^2(5x)cot^5(5x) ?

The integral of csc^2(5x)cot^5(5x) is -1/30 cot^6(5x)+C

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integral of csc^2(5x)cot^5(5x)

\int csc^{2} (5 x) cot^{5} (5 x) d x

- \frac{1}{30} cot^{6} (5 x) + C

Solution steps

\int csc^{2} (5 x) cot^{5} (5 x) d x

Apply u-substitution

: \int csc^{2} (u) cot^{5} (u) \frac{1}{5} d u

= \int csc^{2} (u) cot^{5} (u) \frac{1}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int csc^{2} (u) cot^{5} (u) d u

Rewrite using trig identities

= \frac{1}{5} \cdot \int \frac{cos ^{5} ( u )}{sin ^{7} ( u )} d u

Apply Trigonometric Substitution

: \int - v^{5} d v

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

Take the constant out:

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

= \frac{1}{5} (- \int v^{5} d v)

Apply the Power Rule

: \frac{v ^{6}}{6}

= \frac{1}{5} (- \frac{v ^{6}}{6})

Substitute back

= \frac{1}{5} (- \frac{cot ^{6} ( 5 x )}{6})

Simplify

\frac{1}{5} (- \frac{cot ^{6} ( 5 x )}{6}) : - \frac{1}{30} cot^{6} (5 x)

= - \frac{1}{30} cot^{6} (5 x)

Add a constant to the solution

= - \frac{1}{30} cot^{6} (5 x) + C

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What is the integral of csc^2(5x)cot^5(5x) ?
The integral of csc^2(5x)cot^5(5x) is -1/30 cot^6(5x)+C