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) cos (x) d x

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

Solution steps

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

Take the constant out:

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

= - \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

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