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

The integral of sqrt(sin(2x)) is -\E(2*1/2 \vert (pi/2-2x))+C

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integral of sqrt(sin(2x))

\int sin (2 x) d x

- E (2 \cdot \frac{1}{2} ∣ (\frac{π}{2} - 2 x)) + C

Solution steps

\int sin (2 x) d x

Apply u-substitution

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

= \int 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 sin (u) d u

Use the nonelementary integral:

\int sin (x) d x = - 2 E (\frac{1}{2} (\frac{π}{2} - x) ∣ 2)

= \frac{1}{2} (- 2 E (\frac{1}{2} (\frac{π}{2} - u) ∣ 2))

Substitute back

u = 2 x

= \frac{1}{2} (- 2 E (\frac{1}{2} (\frac{π}{2} - 2 x) ∣ 2))

Simplify

\frac{1}{2} (- 2 E (\frac{1}{2} (\frac{π}{2} - 2 x) ∣ 2)) : - E (2 \cdot \frac{1}{2} ∣ (\frac{π}{2} - 2 x))

= - E (2 \cdot \frac{1}{2} ∣ (\frac{π}{2} - 2 x))

Add a constant to the solution

= - E (2 \cdot \frac{1}{2} ∣ (\frac{π}{2} - 2 x)) + C

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What is the integral of sqrt(sin(2x)) ?
The integral of sqrt(sin(2x)) is -\E(2*1/2 \vert (pi/2-2x))+C