What is the integral of 1/(sin^2(a)x) ?

The integral of 1/(sin^2(a)x) is csc^2(a)ln|x|+C

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integral of 1/(sin^2(a)x)

\int \frac{1}{sin ^{2} ( a ) x} d x

csc^{2} (a) ln ∣ x ∣ + C

Solution steps

\int \frac{1}{sin ^{2} ( a ) x} d x

Take the constant out:

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

= \frac{1}{sin ^{2} ( a )} \cdot \int \frac{1}{x} d x

Use the common integral:

\int \frac{1}{x} d x = ln (∣ x ∣)

= \frac{1}{sin ^{2} ( a )} ln ∣ x ∣

Use the basic trigonometric identity:

\frac{1}{sin ( x )} = csc (x)

= csc^{2} (a) ln ∣ x ∣

Add a constant to the solution

= csc^{2} (a) ln ∣ x ∣ + C

What is the integral of 1/(sin^2(a)x) ?
The integral of 1/(sin^2(a)x) is csc^2(a)ln|x|+C