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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int - u^{2} d u

= 2 \cdot \int - u^{2} d u

Take the constant out:

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

= 2 (- \int u^{2} d u)

Apply the Power Rule

: \frac{u ^{3}}{3}

= 2 (- \frac{u ^{3}}{3})

Substitute back

u = cos (2 x)

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

Simplify

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

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

Add a constant to the solution

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

\int \frac{1}{( x ^{2} + 3 x + 5 ) ^{3}} d x

\int \frac{2}{3 x - 7} d x

x \to - 6 + lim (\frac{2 x}{6 + x})

\frac{\partial}{\partial x} (\frac{( 1 + y )}{( 1 + x )})

2 e^{t}

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