Calculus Examples

$\int \frac{x^{3} + x}{x} d x$

Step 1

Split the fraction into multiple fractions.

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Step 1.1

Factor $x$ out of $x^{3} + x$ .

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Step 1.1.1

Factor $x$ out of $x^{3}$ .

$\int \frac{x \cdot x^{2} + x}{x} d x$

Step 1.1.2

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

$\int \frac{x \cdot x^{2} + x^{1}}{x} d x$

Step 1.1.3

Factor $x$ out of $x^{1}$ .

$\int \frac{x \cdot x^{2} + x \cdot 1}{x} d x$

Step 1.1.4

Factor $x$ out of $x \cdot x^{2} + x \cdot 1$ .

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

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Divide $x^{2} + 1$ by $1$ .

$\int x^{2} + 1 d x$

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

$\frac{1}{3} x^{3} + C + \int d x$

Step 4

Apply the constant rule.

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

Step 5

Simplify.

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