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) 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

1 - x^{2}, 0

\frac{d}{d x} (4 x^{3} - 6 x^{2} - 72 x)

\int (\frac{1}{x ^{7}}) d x

\int cos^{4} (7 x + 7) d x

y = 5 y^{2} (4 y^{2} + 7)^{\frac{6}{5}}

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