Calculus Examples

$x^{2} y$

Step 1

Write $x^{2} y$ as a function.

$f (x) = x^{2} y$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int x^{2} y d x$

Step 4

Since $y$ is constant with respect to $x$ , move $y$ out of the integral.

$y \int x^{2} d x$

Step 5

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$y (\frac{1}{3} x^{3} + C)$

Step 6

Simplify the answer.

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Step 6.1

Rewrite $y (\frac{1}{3} x^{3} + C)$ as $y \frac{1}{3} x^{3} + C$ .

$y \frac{1}{3} x^{3} + C$

Step 6.2

Simplify.

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Step 6.2.1

Combine $y$ and $\frac{1}{3}$ .

$\frac{y}{3} x^{3} + C$

Step 6.2.2

Combine $\frac{y}{3}$ and $x^{3}$ .

$\frac{y x^{3}}{3} + C$

$\frac{1}{3} y x^{3} + C$

Step 7

The answer is the antiderivative of the function $f (x) = x^{2} y$ .

$F (x) =$ $\frac{1}{3} y x^{3} + C$