What is the integral of sin(2x)sqrt(5+sin^2(x)) ?

The integral of sin(2x)sqrt(5+sin^2(x)) is 1/(3sqrt(2))(11-cos(2x))^{3/2}+C

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integral of sin(2x)sqrt(5+sin^2(x))

\int sin (2 x) 5 + sin^{2} (x) d x

\frac{1}{3 2} (11 - cos (2 x))^{\frac{3}{2}} + C

Solution steps

\int sin (2 x) 5 + sin^{2} (x) d x

Rewrite using trig identities

= \int 5 + sin^{2} (x) \cdot 2 cos (x) sin (x) d x

Take the constant out:

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

= 2 \cdot \int 5 + sin^{2} (x) cos (x) sin (x) d x

Rewrite using trig identities

= 2 \cdot \int \frac{1}{2 2} sin (2 x) 11 - cos (2 x) d x

Take the constant out:

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

= 2 \cdot \frac{1}{2 2} \cdot \int sin (2 x) 11 - cos (2 x) d x

Apply u-substitution

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

Take the constant out:

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

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

Apply the Power Rule

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

Substitute back

u = 11 - cos (2 x)

= 2 \cdot \frac{1}{2 2} \cdot \frac{1}{2} \cdot \frac{2}{3} (11 - cos (2 x))^{\frac{3}{2}}

Simplify

2 \cdot \frac{1}{2 2} \cdot \frac{1}{2} \cdot \frac{2}{3} (11 - cos (2 x))^{\frac{3}{2}} : \frac{1}{3 2} (11 - cos (2 x))^{\frac{3}{2}}

= \frac{1}{3 2} (11 - cos (2 x))^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{3 2} (11 - cos (2 x))^{\frac{3}{2}} + C

\frac{d}{d x} (a e^{- 5 x} + b e^{- 3 x} + c x e^{- x})

\int_{0}^{2} 4.0553 e^{1.8 t} d t

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

t a y l or \frac{1}{3 - 2 x}

\frac{d ^{4}}{d x ^{4}} (\frac{1}{x})

What is the integral of sin(2x)sqrt(5+sin^2(x)) ?
The integral of sin(2x)sqrt(5+sin^2(x)) is 1/(3sqrt(2))(11-cos(2x))^{3/2}+C