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

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

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

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

- cos^{3} (t) + C

Solution steps

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

Take the constant out:

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

= 3 \cdot \int sin (t) cos^{2} (t) d t

Apply u-substitution

: \int - u^{2} d u

= 3 \cdot \int - u^{2} d u

Take the constant out:

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

= 3 (- \int u^{2} d u)

Apply the Power Rule

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

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

Substitute back

u = cos (t)

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

Simplify

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

= - cos^{3} (t)

Add a constant to the solution

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

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