What is the integral of sqrt(1+2sin(3x))cos(3x) ?

The integral of sqrt(1+2sin(3x))cos(3x) is 1/9 (1+2sin(3x))^{3/2}+C

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integral of sqrt(1+2sin(3x))cos(3x)

\int 1 + 2 sin (3 x) cos (3 x) d x

\frac{1}{9} (1 + 2 sin (3 x))^{\frac{3}{2}} + C

Solution steps

\int 1 + 2 sin (3 x) cos (3 x) d x

Apply u-substitution

: \int 1 + 2 sin (u) cos (u) \frac{1}{3} d u

= \int 1 + 2 sin (u) cos (u) \frac{1}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int 1 + 2 sin (u) cos (u) d u

Apply u-substitution

: \int \frac{v}{2} d v

= \frac{1}{3} \cdot \int \frac{v}{2} d v

Take the constant out:

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

= \frac{1}{3} \cdot \frac{1}{2} \cdot \int v d v

Apply the Power Rule

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

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

Substitute back

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

Simplify

\frac{1}{3} \cdot \frac{1}{2} \cdot \frac{2}{3} (1 + 2 sin (3 x))^{\frac{3}{2}} : \frac{1}{9} (1 + 2 sin (3 x))^{\frac{3}{2}}

= \frac{1}{9} (1 + 2 sin (3 x))^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{9} (1 + 2 sin (3 x))^{\frac{3}{2}} + C

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What is the integral of sqrt(1+2sin(3x))cos(3x) ?
The integral of sqrt(1+2sin(3x))cos(3x) is 1/9 (1+2sin(3x))^{3/2}+C