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

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

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integral of (2x+1)cot^2(x)

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

- x^{2} - 2 x cot (x) + 2 ln ∣ sin (x) ∣ - cot (x) - x + C

Solution steps

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

Rewrite using trig identities

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

Expand

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

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

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

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

= - x^{2} - 2 x cot (x) + 2 ln ∣ sin (x) ∣ - cot (x) - x

Add a constant to the solution

= - x^{2} - 2 x cot (x) + 2 ln ∣ sin (x) ∣ - cot (x) - x + C

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