Calculus Examples

$(x^{2} - 1) d x - (y^{2} + x) d y$

Step 1

Split the single integral into multiple integrals.

$\int (x^{2} - 1) d x d x + \int - (y^{2} + x) d y d x$

Step 2

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

$d \int (x^{2} - 1) x d x + \int - (y^{2} + x) d y d x$

Step 3

Multiply $(x^{2} - 1) x$ .

$d \int x^{2} x - 1 x d x + \int - (y^{2} + x) d y d x$

Step 4

Simplify.

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Step 4.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 4.1.1

Multiply $x^{2}$ by $x$ .

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Step 4.1.1.1

Raise $x$ to the power of $1$ .

$d \int x^{2} x^{1} - 1 x d x + \int - (y^{2} + x) d y d x$

Step 4.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$d \int x^{2 + 1} - 1 x d x + \int - (y^{2} + x) d y d x$

Step 4.1.2

Add $2$ and $1$ .

$d \int x^{3} - 1 x d x + \int - (y^{2} + x) d y d x$

Step 4.2

Rewrite $- 1 x$ as $- x$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 8

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

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

Step 9

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

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

Step 10

Split the single integral into multiple integrals.

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

Step 11

Apply the constant rule.

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

Step 12

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

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

Step 13

Simplify.

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Step 13.1

Simplify.

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Step 13.1.1

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

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

Step 13.1.2

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

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

Step 13.1.3

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

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

Step 13.2

Simplify.

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

Step 13.3

Simplify.

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Step 13.3.1

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

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

Step 13.3.2

To write $- d y (y^{2} x + \frac{x^{2}}{2})$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 13.3.3

Combine $- d y (y^{2} x + \frac{x^{2}}{2})$ and $\frac{4}{4}$ .

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

Step 13.3.4

Combine the numerators over the common denominator.

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

Step 13.3.5

Multiply $4$ by $- 1$ .

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

Step 14

Reorder terms.

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