What is the integral of cos^3(x)sin(2x) ?

The integral of cos^3(x)sin(2x) is -2/5 cos^5(x)+C

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integral of cos^3(x)sin(2x)

\int cos^{3} (x) sin (2 x) d x

- \frac{2}{5} cos^{5} (x) + C

Solution steps

\int cos^{3} (x) sin (2 x) d x

Rewrite using trig identities

= \int 2 cos^{4} (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 cos^{4} (x) sin (x) d x

Apply u-substitution

: \int - u^{4} d u

= 2 \cdot \int - u^{4} d u

Take the constant out:

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

= 2 (- \int u^{4} d u)

Apply the Power Rule

: \frac{u ^{5}}{5}

= 2 (- \frac{u ^{5}}{5})

Substitute back

u = cos (x)

= 2 (- \frac{cos ^{5} ( x )}{5})

Simplify

2 (- \frac{cos ^{5} ( x )}{5}) : - \frac{2}{5} cos^{5} (x)

= - \frac{2}{5} cos^{5} (x)

Add a constant to the solution

= - \frac{2}{5} cos^{5} (x) + C

f (x) = x^{4} - 5 x^{3} + 2, at x = 2

\frac{d}{d x} (5 ln (x^{3}))

n = 0 \sum \infty \frac{1}{( 2 n ! )}

\int x cos (mπ x) d x

\frac{\partial}{\partial x} (9 x e^{6 x y})

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