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

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

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

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

\frac{1}{3} sin^{\frac{3}{2}} (2 x) + C

Solution steps

\int sin (2 x) cos (2 x) d x

Apply u-substitution

= \int sin (u) cos (u) \frac{1}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int sin (u) cos (u) d u

Apply u-substitution

= \frac{1}{2} \cdot \int v d v

Apply the Power Rule

= \frac{1}{2} \cdot \frac{2}{3} v^{\frac{3}{2}}

Substitute back

= \frac{1}{2} \cdot \frac{2}{3} sin^{\frac{3}{2}} (2 x)

Simplify

\frac{1}{2} \cdot \frac{2}{3} sin^{\frac{3}{2}} (2 x) : \frac{1}{3} sin^{\frac{3}{2}} (2 x)

= \frac{1}{3} sin^{\frac{3}{2}} (2 x)

Add a constant to the solution

= \frac{1}{3} sin^{\frac{3}{2}} (2 x) + C

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