What is the integral of cot^3(7x) ?

The integral of cot^3(7x) is 1/7 (-(cot^2(7x))/2-ln|sin(7x)|)+C

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integral of cot^3(7x)

\int cot^{3} (7 x) d x

\frac{1}{7} (- \frac{cot ^{2} ( 7 x )}{2} - ln ∣ sin (7 x) ∣) + C

Solution steps

\int cot^{3} (7 x) d x

Apply u-substitution

: \int cot^{3} (u) \frac{1}{7} d u

= \int cot^{3} (u) \frac{1}{7} d u

Take the constant out:

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

= \frac{1}{7} \cdot \int cot^{3} (u) d u

Apply Integral Reduction

= \frac{1}{7} (- \frac{cot ^{2} ( u )}{2} - \int cot (u) d u)

\int cot (u) d u = ln ∣ sin (u) ∣

= \frac{1}{7} (- \frac{cot ^{2} ( u )}{2} - ln ∣ sin (u) ∣)

Substitute back

u = 7 x

= \frac{1}{7} (- \frac{cot ^{2} ( 7 x )}{2} - ln ∣ sin (7 x) ∣)

Add a constant to the solution

= \frac{1}{7} (- \frac{cot ^{2} ( 7 x )}{2} - ln ∣ sin (7 x) ∣) + C

R (a) = (5 a + 1)^{2}

\frac{d}{d x} (y \cdot e^{x})

\frac{d}{d x} (\frac{54}{( 1 + 3 x ) ^{3}})

y^{^{'}} = (1 - 2 t) y

\frac{d}{d x} (x cot (\frac{5}{x ^{2}}))

What is the integral of cot^3(7x) ?
The integral of cot^3(7x) is 1/7 (-(cot^2(7x))/2-ln|sin(7x)|)+C