What is the integral of xcsc^2(ax) ?

The integral of xcsc^2(ax) is -1/a xcot(ax)+1/(a^2)ln|sin(ax)|+C

Popular >

integral of xcsc^2(ax)

\int x csc^{2} (a x) d x

- \frac{1}{a} x cot (a x) + \frac{1}{a ^{2}} ln ∣ sin (a x) ∣ + C

Solution steps

\int x csc^{2} (a x) d x

Apply Integration By Parts

: - \frac{1}{a} x cot (a x) - \int - \frac{1}{a} cot (a x) d x

= - \frac{1}{a} x cot (a x) - \int - \frac{1}{a} cot (a x) d x

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

= - \frac{1}{a} x cot (a x) - (- \frac{1}{a ^{2}} ln ∣ sin (a x) ∣)

Simplify

= - \frac{1}{a} x cot (a x) + \frac{1}{a ^{2}} ln ∣ sin (a x) ∣

Add a constant to the solution

= - \frac{1}{a} x cot (a x) + \frac{1}{a ^{2}} ln ∣ sin (a x) ∣ + C

\int \frac{tan ( x )}{1 + tan ( x )} d x

\int \frac{( cos ( t ) ) ^{5}}{sin ( t )} d t

\int \frac{15 e ^{5 w}}{e ^{5 w} + 2} d w

\int \frac{e ^{t}}{e ^{t} + 1} d t

\int - 8 cos (x) d x

What is the integral of xcsc^2(ax) ?
The integral of xcsc^2(ax) is -1/a xcot(ax)+1/(a^2)ln|sin(ax)|+C