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

\int \frac{( e ^{2 x} - 1 )}{e ^{x}} d x

x \to 3 lim (2 x)

y^{^{'}} = cos (x - y)

\frac{9 e ^{x} x ^{\frac{7}{2}}}{2} + e^{x} x^{\frac{9}{2}}

e^{x}, e^{- 2 x}, ln (4)

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