What is the integral of (sin^2(x))/(sqrt(1-cos(x))) ?

The integral of (sin^2(x))/(sqrt(1-cos(x))) is -(4sqrt(2))/3 cos^3(x/2)+C

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integral of (sin^2(x))/(sqrt(1-cos(x)))

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

- \frac{4 2}{3} cos^{3} (\frac{x}{2}) + C

Solution steps

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

Apply u-substitution

: \int \frac{4 2 u}{( 1 + u ^{2} ) ^{\frac{5}{2}}} d u

= \int \frac{4 2 u}{( 1 + u ^{2} ) ^{\frac{5}{2}}} d u

Take the constant out:

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

= 42 \cdot \int \frac{u}{( 1 + u ^{2} ) ^{\frac{5}{2}}} d u

Apply u-substitution

: \int \frac{1}{2 v ^{\frac{5}{2}}} d v

= 42 \cdot \int \frac{1}{2 v ^{\frac{5}{2}}} d v

Take the constant out:

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

= 42 \frac{1}{2} \cdot \int \frac{1}{v ^{\frac{5}{2}}} d v

Apply the Power Rule

: - \frac{2}{3 v ^{\frac{3}{2}}}

= 42 \frac{1}{2} (- \frac{2}{3 v ^{\frac{3}{2}}})

Substitute back

= 42 \frac{1}{2} - \frac{2}{3 ( 1 + tan ^{2} ( \frac{x}{2} ) ) ^{\frac{3}{2}}}

Simplify

42 \frac{1}{2} - \frac{2}{3 ( 1 + tan ^{2} ( \frac{x}{2} ) ) ^{\frac{3}{2}}} : - \frac{4 2}{3} cos^{3} (\frac{x}{2})

= - \frac{4 2}{3} cos^{3} (\frac{x}{2})

Add a constant to the solution

= - \frac{4 2}{3} cos^{3} (\frac{x}{2}) + C

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What is the integral of (sin^2(x))/(sqrt(1-cos(x))) ?
The integral of (sin^2(x))/(sqrt(1-cos(x))) is -(4sqrt(2))/3 cos^3(x/2)+C