What is the integral of sin(2t)cos(t) ?

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

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integral of sin(2t)cos(t)

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

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

Solution steps

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

Rewrite using trig identities

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

Take the constant out:

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

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

Apply u-substitution

= 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

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

Substitute back

u = cos (t)

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

Simplify

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

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

Add a constant to the solution

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

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