What is the integral of (cot(x)+3)^2 ?

The integral of (cot(x)+3)^2 is 8x-cot(x)+6ln|sin(x)|+C

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integral of (cot(x)+3)^2

\int (cot (x) + 3)^{2} d x

8 x - cot (x) + 6 ln ∣ sin (x) ∣ + C

Solution steps

\int (cot (x) + 3)^{2} d x

Rewrite using trig identities

= \int \frac{( cos ( x ) + 3 sin ( x ) ) ^{2}}{sin ^{2} ( x )} d x

Expand

\frac{( cos ( x ) + 3 sin ( x ) ) ^{2}}{sin ^{2} ( x )} : \frac{cos ^{2} ( x )}{sin ^{2} ( x )} + \frac{6 cos ( x )}{sin ( x )} + 9

= \int \frac{cos ^{2} ( x )}{sin ^{2} ( x )} + \frac{6 cos ( x )}{sin ( x )} + 9 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 \frac{cos ^{2} ( x )}{sin ^{2} ( x )} d x + \int \frac{6 cos ( x )}{sin ( x )} d x + \int 9 d x

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

\int \frac{6 cos ( x )}{sin ( x )} d x = 6 ln ∣ sin (x) ∣

\int 9 d x = 9 x

= - cot (x) - x + 6 ln ∣ sin (x) ∣ + 9 x

Simplify

= 8 x - cot (x) + 6 ln ∣ sin (x) ∣

Add a constant to the solution

= 8 x - cot (x) + 6 ln ∣ sin (x) ∣ + C

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What is the integral of (cot(x)+3)^2 ?
The integral of (cot(x)+3)^2 is 8x-cot(x)+6ln|sin(x)|+C