What is the integral of (x-3)csc^2(x) ?

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

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integral of (x-3)csc^2(x)

\int (x - 3) csc^{2} (x) d x

- x cot (x) + ln ∣ sin (x) ∣ + 3 cot (x) + C

Solution steps

\int (x - 3) csc^{2} (x) d x

Rewrite using trig identities

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

Expand

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

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

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

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

= - x cot (x) + ln ∣ sin (x) ∣ - (- 3 cot (x))

Simplify

= - x cot (x) + ln ∣ sin (x) ∣ + 3 cot (x)

Add a constant to the solution

= - x cot (x) + ln ∣ sin (x) ∣ + 3 cot (x) + C

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