What is the integral of (1-cos(x))^{3/2} ?

The integral of (1-cos(x))^{3/2} is 2*2^{1/2}(-2cos(x/2)+2/3 cos^3(x/2))+C

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integral of (1-cos(x))^{3/2}

\int (1 - cos (x))^{\frac{3}{2}} d x

2 \cdot 2^{\frac{1}{2}} (- 2 cos (\frac{x}{2}) + \frac{2}{3} cos^{3} (\frac{x}{2})) + C

Solution steps

\int (1 - cos (x))^{\frac{3}{2}} d x

Apply u-substitution

: \int \frac{4 \cdot 2 ^{\frac{1}{2}} u ^{3}}{( 1 + u ^{2} ) ^{\frac{5}{2}}} d u

= \int \frac{4 \cdot 2 ^{\frac{1}{2}} u ^{3}}{( 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

= 4 \cdot 2^{\frac{1}{2}} \cdot \int \frac{u ^{3}}{( 1 + u ^{2} ) ^{\frac{5}{2}}} d u

Apply u-substitution

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

= 4 \cdot 2^{\frac{1}{2}} \cdot \int \frac{v - 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

= 4 \cdot 2^{\frac{1}{2}} \cdot \frac{1}{2} \cdot \int \frac{v - 1}{v ^{\frac{5}{2}}} d v

Expand

\frac{v - 1}{v ^{\frac{5}{2}}} : \frac{1}{v ^{\frac{3}{2}}} - \frac{1}{v ^{\frac{5}{2}}}

= 4 \cdot 2^{\frac{1}{2}} \cdot \frac{1}{2} \cdot \int \frac{1}{v ^{\frac{3}{2}}} - \frac{1}{v ^{\frac{5}{2}}} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 4 \cdot 2^{\frac{1}{2}} \cdot \frac{1}{2} (\int \frac{1}{v ^{\frac{3}{2}}} d v - \int \frac{1}{v ^{\frac{5}{2}}} d v)

\int \frac{1}{v ^{\frac{3}{2}}} d v = - \frac{2}{v ^{\frac{1}{2}}}

\int \frac{1}{v ^{\frac{5}{2}}} d v = - \frac{2}{3 v ^{\frac{3}{2}}}

= 4 \cdot 2^{\frac{1}{2}} \cdot \frac{1}{2} (- \frac{2}{v ^{\frac{1}{2}}} - (- \frac{2}{3 v ^{\frac{3}{2}}}))

Substitute back

= 4 \cdot 2^{\frac{1}{2}} \cdot \frac{1}{2} - \frac{2}{( 1 + tan ^{2} ( \frac{x}{2} ) ) ^{\frac{1}{2}}} - - \frac{2}{3 ( 1 + tan ^{2} ( \frac{x}{2} ) ) ^{\frac{3}{2}}}

Simplify

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

= 2 \cdot 2^{\frac{1}{2}} (- 2 cos (\frac{x}{2}) + \frac{2}{3} cos^{3} (\frac{x}{2}))

Add a constant to the solution

= 2 \cdot 2^{\frac{1}{2}} (- 2 cos (\frac{x}{2}) + \frac{2}{3} cos^{3} (\frac{x}{2})) + C

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What is the integral of (1-cos(x))^{3/2} ?
The integral of (1-cos(x))^{3/2} is 2*2^{1/2}(-2cos(x/2)+2/3 cos^3(x/2))+C