What is the integral of (cos(x/3))/(sin^4(x/3)) ?

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

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integral of (cos(x/3))/(sin^4(x/3))

\int \frac{cos ( \frac{x}{3} )}{sin ^{4} ( \frac{x}{3} )} d x

- csc^{3} (\frac{x}{3}) + C

Solution steps

\int \frac{cos ( \frac{x}{3} )}{sin ^{4} ( \frac{x}{3} )} d x

Apply u-substitution

: \int \frac{3 cos ( u )}{sin ^{4} ( u )} d u

= \int \frac{3 cos ( u )}{sin ^{4} ( u )} d u

Take the constant out:

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

= 3 \cdot \int \frac{cos ( u )}{sin ^{4} ( u )} d u

Apply u-substitution

: \int \frac{1}{v ^{4}} d v

= 3 \cdot \int \frac{1}{v ^{4}} d v

Apply the Power Rule

: - \frac{1}{3 v ^{3}}

= 3 (- \frac{1}{3 v ^{3}})

Substitute back

= 3 (- \frac{1}{3 sin ^{3} ( \frac{x}{3} )})

Simplify

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

= - csc^{3} (\frac{x}{3})

Add a constant to the solution

= - csc^{3} (\frac{x}{3}) + C

\int e^{7 x} cos (3 x) d x

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

\frac{1}{4} y^{^{''}} + y^{^{'}} + y = x^{2} - 4 x

y \cdot y^{^{'}} + 36 x = 0

y = x, y = \frac{x}{2}

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