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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int - u d u

= \int - u d u

Take the constant out:

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

= - \int u d u

Apply the Power Rule

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

= - \frac{u ^{2}}{2}

Substitute back

u = cos^{2} (x)

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

Simplify

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

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

Add a constant to the solution

= - \frac{1}{2} cos^{4} (x) + C

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

\int 20 cot^{4} (x) d x

\frac{\partial}{\partial x} (10 x^{3} + 3 x^{2} y - 3 y^{5})

y^{^{''}} - 11 y^{^{'}} + 30 y = 20 e^{t}

\int_{0}^{2} (sin (x) + e^{x}) d x

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