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

The integral of cot(2x)(cot(2x)+3csc(2x)) is -x-1/2 cot(2x)+3/4 (-cot(x)-tan(x))+C

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

\int cot (2 x) (cot (2 x) + 3 csc (2 x)) d x

- x - \frac{1}{2} cot (2 x) + \frac{3}{4} (- cot (x) - tan (x)) + C

Solution steps

\int cot (2 x) (cot (2 x) + 3 csc (2 x)) d x

Rewrite using trig identities

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

Expand

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

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

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

\int \frac{3 cos ( 2 x )}{sin ^{2} ( 2 x )} d x = \frac{3}{4} (- cot (x) - tan (x))

= - x - \frac{1}{2} cot (2 x) + \frac{3}{4} (- cot (x) - tan (x))

Add a constant to the solution

= - x - \frac{1}{2} cot (2 x) + \frac{3}{4} (- cot (x) - tan (x)) + C

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What is the integral of cot(2x)(cot(2x)+3csc(2x)) ?
The integral of cot(2x)(cot(2x)+3csc(2x)) is -x-1/2 cot(2x)+3/4 (-cot(x)-tan(x))+C