What is the integral of 12sin^2(t)cos(t) ?

The integral of 12sin^2(t)cos(t) is 4sin^3(t)+C

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integral of 12sin^2(t)cos(t)

\int 12 sin^{2} (t) cos (t) d t

4 sin^{3} (t) + C

Solution steps

\int 12 sin^{2} (t) cos (t) d t

Take the constant out:

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

= 12 \cdot \int sin^{2} (t) cos (t) d t

Apply u-substitution

= 12 \cdot \int u^{2} d u

Apply the Power Rule

= 12 \cdot \frac{u ^{3}}{3}

Substitute back

u = sin (t)

= 12 \cdot \frac{sin ^{3} ( t )}{3}

Simplify

12 \cdot \frac{sin ^{3} ( t )}{3} : 4 sin^{3} (t)

= 4 sin^{3} (t)

Add a constant to the solution

= 4 sin^{3} (t) + C

What is the integral of 12sin^2(t)cos(t) ?
The integral of 12sin^2(t)cos(t) is 4sin^3(t)+C