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

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

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

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

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

Solution steps

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

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