What is the integral of x(3-x^2)^2 ?

The integral of x(3-x^2)^2 is -1/6 (3-x^2)^3+C

Popular >

integral of x(3-x^2)^2

\int x (3 - x^{2})^{2} d x

- \frac{1}{6} (3 - x^{2})^{3} + C

Solution steps

\int x (3 - x^{2})^{2} d x

Apply u-substitution

: \int - \frac{u ^{2}}{2} d u

= \int - \frac{u ^{2}}{2} d u

Take the constant out:

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

= - \frac{1}{2} \cdot \int u^{2} d u

Apply the Power Rule

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

= - \frac{1}{2} \cdot \frac{u ^{3}}{3}

Substitute back

u = 3 - x^{2}

= - \frac{1}{2} \cdot \frac{( 3 - x ^{2} ) ^{3}}{3}

Simplify

- \frac{1}{2} \cdot \frac{( 3 - x ^{2} ) ^{3}}{3} : - \frac{1}{6} (3 - x^{2})^{3}

= - \frac{1}{6} (3 - x^{2})^{3}

Add a constant to the solution

= - \frac{1}{6} (3 - x^{2})^{3} + C

x \to 3 lim (\frac{1}{x - 3} + 2)

4 t \frac{d y}{d t} + y = t^{2}

f (x) = 72 x^{3} (3 x^{4} + 7)^{5}

\frac{d y}{d x} = \frac{y}{2 x}

f (x) = \frac{x}{4 x + 5}

What is the integral of x(3-x^2)^2 ?
The integral of x(3-x^2)^2 is -1/6 (3-x^2)^3+C