Calculus Examples

$f (x) = 3 x^{3} + 3 y^{3} - 6 x y$

Step 1

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 2

Set up the integral to solve.

$F (x) = \int 3 x^{3} + 3 y^{3} - 6 x y d x$

Step 3

Split the single integral into multiple integrals.

$\int 3 x^{3} d x + \int 3 y^{3} d x + \int - 6 x y d x$

Step 4

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int x^{3} d x + \int 3 y^{3} d x + \int - 6 x y d x$

Step 5

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$3 (\frac{1}{4} x^{4} + C) + \int 3 y^{3} d x + \int - 6 x y d x$

Step 6

Apply the constant rule.

$3 (\frac{1}{4} x^{4} + C) + 3 y^{3} x + C + \int - 6 x y d x$

Step 7

Combine $\frac{1}{4}$ and $x^{4}$ .

$3 (\frac{x^{4}}{4} + C) + 3 y^{3} x + C + \int - 6 x y d x$

Step 8

Since $- 6 y$ is constant with respect to $x$ , move $- 6 y$ out of the integral.

$3 (\frac{x^{4}}{4} + C) + 3 y^{3} x + C - 6 y \int x d x$

Step 9

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

$3 (\frac{x^{4}}{4} + C) + 3 y^{3} x + C - 6 y (\frac{1}{2} x^{2} + C)$

Step 10

Simplify.

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Step 10.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$3 (\frac{x^{4}}{4} + C) + 3 y^{3} x + C - 6 y (\frac{x^{2}}{2} + C)$

Step 10.2

Simplify.

$\frac{3 x^{4}}{4} + 3 y^{3} x - 6 y \frac{x^{2}}{2} + C$

Step 10.3

Simplify.

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Step 10.3.1

Combine $- 6$ and $\frac{x^{2}}{2}$ .

$\frac{3 x^{4}}{4} + 3 y^{3} x + y \frac{- 6 x^{2}}{2} + C$

Step 10.3.2

Combine $y$ and $\frac{- 6 x^{2}}{2}$ .

$\frac{3 x^{4}}{4} + 3 y^{3} x + \frac{y (- 6 x^{2})}{2} + C$

Step 10.3.3

Cancel the common factor of $- 6$ and $2$ .

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Step 10.3.3.1

Factor $2$ out of $y (- 6 x^{2})$ .

$\frac{3 x^{4}}{4} + 3 y^{3} x + \frac{2 (y (- 3 x^{2}))}{2} + C$

Step 10.3.3.2

Cancel the common factors.

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Step 10.3.3.2.1

Factor $2$ out of $2$ .

$\frac{3 x^{4}}{4} + 3 y^{3} x + \frac{2 (y (- 3 x^{2}))}{2 (1)} + C$

Step 10.3.3.2.2

Cancel the common factor.

$\frac{3 x^{4}}{4} + 3 y^{3} x + \frac{2 (y (- 3 x^{2}))}{2 \cdot 1} + C$

Step 10.3.3.2.3

Rewrite the expression.

$\frac{3 x^{4}}{4} + 3 y^{3} x + \frac{y (- 3 x^{2})}{1} + C$

Step 10.3.3.2.4

Divide $y (- 3 x^{2})$ by $1$ .

$\frac{3 x^{4}}{4} + 3 y^{3} x + y (- 3 x^{2}) + C$

Step 10.4

Reorder terms.

$\frac{3}{4} x^{4} + 3 y^{3} x + y (- 3 x^{2}) + C$

Step 11

The answer is the antiderivative of the function $f (x) = 3 x^{3} + 3 y^{3} - 6 x y$ .

$F (x) =$ $\frac{3}{4} x^{4} + 3 y^{3} x + y (- 3 x^{2}) + C$