What is the integral of (1-cot(5x))^2 ?

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

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integral of (1-cot(5x))^2

\int (1 - cot (5 x))^{2} d x

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

Solution steps

\int (1 - cot (5 x))^{2} d x

Rewrite using trig identities

= \int \frac{( sin ( 5 x ) - cos ( 5 x ) ) ^{2}}{sin ^{2} ( 5 x )} d x

Expand

\frac{( sin ( 5 x ) - cos ( 5 x ) ) ^{2}}{sin ^{2} ( 5 x )} : 1 - \frac{2 cos ( 5 x )}{sin ( 5 x )} + \frac{cos ^{2} ( 5 x )}{sin ^{2} ( 5 x )}

= \int 1 - \frac{2 cos ( 5 x )}{sin ( 5 x )} + \frac{cos ^{2} ( 5 x )}{sin ^{2} ( 5 x )} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d x - \int \frac{2 cos ( 5 x )}{sin ( 5 x )} d x + \int \frac{cos ^{2} ( 5 x )}{sin ^{2} ( 5 x )} d x

\int 1 d x = x

\int \frac{2 cos ( 5 x )}{sin ( 5 x )} d x = \frac{2}{5} ln ∣ sin (5 x) ∣

\int \frac{cos ^{2} ( 5 x )}{sin ^{2} ( 5 x )} d x = - x - \frac{1}{5} cot (5 x)

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

Simplify

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

Add a constant to the solution

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

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