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

The integral of (sin^3(x))/(sin^4(x)) is ln|tan(x/2)|+C

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

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

ln tan (\frac{x}{2}) + C

Solution steps

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

Simplify

\frac{sin ^{3} ( x )}{sin ^{4} ( x )} : csc (x)

= \int csc (x) d x

Apply u-substitution

: \int \frac{1}{u} d u

= \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = tan (\frac{x}{2})

= ln tan (\frac{x}{2})

Add a constant to the solution

= ln tan (\frac{x}{2}) + C

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What is the integral of (sin^3(x))/(sin^4(x)) ?
The integral of (sin^3(x))/(sin^4(x)) is ln|tan(x/2)|+C