What is the integral of 4csc(x)(csc(x)-3cot(x)) ?

The integral of 4csc(x)(csc(x)-3cot(x)) is 4(-cot(x)+3csc(x))+C

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integral of 4csc(x)(csc(x)-3cot(x))

\int 4 csc (x) (csc (x) - 3 cot (x)) d x

4 (- cot (x) + 3 csc (x)) + C

Solution steps

\int 4 csc (x) (csc (x) - 3 cot (x)) d x

Take the constant out:

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

= 4 \cdot \int csc (x) (csc (x) - 3 cot (x)) d x

Rewrite using trig identities

= 4 \cdot \int \frac{1 - 3 cos ( x )}{sin ^{2} ( x )} d x

Expand

\frac{1 - 3 cos ( x )}{sin ^{2} ( x )} : \frac{1}{sin ^{2} ( x )} - \frac{3 cos ( x )}{sin ^{2} ( x )}

= 4 \cdot \int \frac{1}{sin ^{2} ( x )} - \frac{3 cos ( 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

= 4 (\int \frac{1}{sin ^{2} ( x )} d x - \int \frac{3 cos ( x )}{sin ^{2} ( x )} d x)

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

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

= 4 (- cot (x) - (- 3 csc (x)))

Simplify

= 4 (- cot (x) + 3 csc (x))

Add a constant to the solution

= 4 (- cot (x) + 3 csc (x)) + C

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