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

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

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

\int sin (2 x) \cdot cos (x) d x

- \frac{2}{3} cos^{3} (x) + C

Solution steps

\int sin (2 x) cos (x) d x

Rewrite using trig identities

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

Apply u-substitution

: \int - u^{2} d u

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

Take the constant out:

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

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

Apply the Power Rule

: \frac{u ^{3}}{3}

= 2 (- \frac{u ^{3}}{3})

Substitute back

u = cos (x)

= 2 (- \frac{cos ^{3} ( x )}{3})

Simplify

2 (- \frac{cos ^{3} ( x )}{3}) : - \frac{2}{3} cos^{3} (x)

= - \frac{2}{3} cos^{3} (x)

Add a constant to the solution

= - \frac{2}{3} cos^{3} (x) + C

\frac{d}{d x} (x^{3} - 3 x^{2} + 2 x)

\frac{\partial}{\partial x} (x^{3} y^{3} + y - z + 3)

\frac{\partial}{\partial x} (x^{7} y^{8} - x^{6} y^{7})

e^{- 2 t} sin (3 t)

\int x csc^{2} (x) d x

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