What is the integral of (cos^2(x)sin(2x))/2 ?

The integral of (cos^2(x)sin(2x))/2 is -1/4 cos^4(x)+C

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integral of (cos^2(x)sin(2x))/2

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int - u d u

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

Take the constant out:

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

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

Apply the Power Rule

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

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

Substitute back

u = cos^{2} (x)

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

Simplify

\frac{1}{2} (- \frac{( cos ^{2} ( x ) ) ^{2}}{2}) : - \frac{1}{4} cos^{4} (x)

= - \frac{1}{4} cos^{4} (x)

Add a constant to the solution

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

\int (y sin (x)) d x

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\int_{0}^{0.2} 8 e^{- 8 x} d x

\int x (e^{x} + e^{x^{2}}) d x

2 x^{2} + 11, 0, - 2, 2

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