What is the integral of sqrt(1-sin(2x)) ?

The integral of sqrt(1-sin(2x)) is sqrt(sin(2x)+1)+C

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integral of sqrt(1-sin(2x))

\int 1 - sin (2 x) d x

sin (2 x) + 1 + C

Solution steps

\int 1 - sin (2 x) d x

Apply u-substitution

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

= \int 1 - 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 1 - sin (u) d u

Apply u-substitution

: \int \frac{1}{v + 1} d v

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

Apply u-substitution

: \int \frac{1}{w} d w

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

Simplify

\frac{1}{w} : \frac{1}{w}

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

Simplify

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

Apply the Power Rule

: 2 w^{\frac{1}{2}}

= \frac{1}{2} \cdot 2 w^{\frac{1}{2}}

Substitute back

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

Simplify

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

= sin (2 x) + 1

Add a constant to the solution

= sin (2 x) + 1 + C

n = 0 \sum \infty (- \frac{10}{11})^{n}

7 t \frac{d y}{d t} + y = 0, y (1) = 1

\int \frac{c}{x} d x

\int 3 x^{2} - 4 x + 5 d x

x \to 0.1 lim (\frac{e ^{5 x} - 1}{x})

What is the integral of sqrt(1-sin(2x)) ?
The integral of sqrt(1-sin(2x)) is sqrt(sin(2x)+1)+C