What is the integral of sin(4x)sqrt(cos(4x)+1) ?

The integral of sin(4x)sqrt(cos(4x)+1) is -1/6 (cos(4x)+1)^{3/2}+C

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integral of sin(4x)sqrt(cos(4x)+1)

\int sin (4 x) cos (4 x) + 1 d x

- \frac{1}{6} (cos (4 x) + 1)^{\frac{3}{2}} + C

Solution steps

\int sin (4 x) cos (4 x) + 1 d x

Apply u-substitution

: \int - \frac{u}{4} d u

= \int - \frac{u}{4} d u

Take the constant out:

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

= - \frac{1}{4} \cdot \int u d u

Apply the Power Rule

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

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

Substitute back

u = cos (4 x) + 1

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

Simplify

- \frac{1}{4} \cdot \frac{2}{3} (cos (4 x) + 1)^{\frac{3}{2}} : - \frac{1}{6} (cos (4 x) + 1)^{\frac{3}{2}}

= - \frac{1}{6} (cos (4 x) + 1)^{\frac{3}{2}}

Add a constant to the solution

= - \frac{1}{6} (cos (4 x) + 1)^{\frac{3}{2}} + C

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What is the integral of sin(4x)sqrt(cos(4x)+1) ?
The integral of sin(4x)sqrt(cos(4x)+1) is -1/6 (cos(4x)+1)^{3/2}+C