What is the integral of (cos^3(x))/(csc(x)) ?

The integral of (cos^3(x))/(csc(x)) is -(cos^4(x))/4+C

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integral of (cos^3(x))/(csc(x))

\int \frac{cos ^{3} ( x )}{csc ( x )} d x

- \frac{cos ^{4} ( x )}{4} + C

Solution steps

\int \frac{cos ^{3} ( x )}{csc ( x )} d x

Use the basic trigonometric identity:

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

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

Apply u-substitution

= \int - u^{3} d u

Take the constant out:

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

= - \int u^{3} d u

Apply the Power Rule

= - \frac{u ^{4}}{4}

Substitute back

u = cos (x)

= - \frac{cos ^{4} ( x )}{4}

Add a constant to the solution

= - \frac{cos ^{4} ( x )}{4} + C

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