What is the integral of cot^3(5x) ?

The integral of cot^3(5x) is 1/5 (-(cot^2(5x))/2-ln|sin(5x)|)+C

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integral of cot^3(5x)

\int cot^{3} (5 x) d x

\frac{1}{5} (- \frac{cot ^{2} ( 5 x )}{2} - ln ∣ sin (5 x) ∣) + C

Solution steps

\int cot^{3} (5 x) d x

Apply u-substitution

: \int cot^{3} (u) \frac{1}{5} d u

= \int cot^{3} (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 cot^{3} (u) d u

Apply Integral Reduction

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

\int cot (u) d u = ln ∣ sin (u) ∣

= \frac{1}{5} (- \frac{cot ^{2} ( u )}{2} - ln ∣ sin (u) ∣)

Substitute back

u = 5 x

= \frac{1}{5} (- \frac{cot ^{2} ( 5 x )}{2} - ln ∣ sin (5 x) ∣)

Add a constant to the solution

= \frac{1}{5} (- \frac{cot ^{2} ( 5 x )}{2} - ln ∣ sin (5 x) ∣) + C

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\int \frac{2 x ^{3} + 7 x ^{2} + 2 x + 9}{2 x + 3} d x

What is the integral of cot^3(5x) ?
The integral of cot^3(5x) is 1/5 (-(cot^2(5x))/2-ln|sin(5x)|)+C