What is the integral of (sin(2x))(cos(2x)+1)^{1/2} ?

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

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integral of (sin(2x))(cos(2x)+1)^{1/2}

\int (sin (2 x)) (cos (2 x) + 1)^{\frac{1}{2}} d x

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

Solution steps

\int sin (2 x) (cos (2 x) + 1)^{\frac{1}{2}} d x

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

= \int 2^{\frac{1}{2}} sin (2 x) cos (x) d x

Take the constant out:

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

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

Rewrite using trig identities

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

Take the constant out:

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

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

Apply u-substitution

: \int - u^{2} d u

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

Take the constant out:

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

= 2^{\frac{1}{2}} \cdot 2 (- \int u^{2} d u)

Apply the Power Rule

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

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

Substitute back

u = cos (x)

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

Simplify

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

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

Add a constant to the solution

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

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