What is the integral of cos(2x)sqrt(sin(2x)) ?

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

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integral of cos(2x)sqrt(sin(2x))

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

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

Solution steps

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

Apply u-substitution

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

= \int cos (u) sin (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 cos (u) sin (u) d u

Apply u-substitution

: \int v d v

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

Apply the Power Rule

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

= \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

\frac{d}{d x} (sec (ln (x)))

\int e^{2 t} sin (t) d t

a^{6} + cos^{6} (x)

\int_{- 3}^{1} (1 + 4 x) d x

x \to 6 lim (2 x - 5)

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