Calculus Examples

$\int \frac{x + 2 x^{2} + 3 x^{3}}{x} d x$

Step 1

Split the fraction into multiple fractions.

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Step 1.1

Factor $x$ out of $x + 2 x^{2} + 3 x^{3}$ .

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Step 1.1.1

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

$\int \frac{x^{1} + 2 x^{2} + 3 x^{3}}{x} d x$

Step 1.1.2

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

$\int \frac{x \cdot 1 + 2 x^{2} + 3 x^{3}}{x} d x$

Step 1.1.3

Factor $x$ out of $2 x^{2}$ .

$\int \frac{x \cdot 1 + x (2 x) + 3 x^{3}}{x} d x$

Step 1.1.4

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

$\int \frac{x \cdot 1 + x (2 x) + x (3 x^{2})}{x} d x$

Step 1.1.5

Factor $x$ out of $x \cdot 1 + x (2 x)$ .

$\int \frac{x \cdot (1 + 2 x) + x (3 x^{2})}{x} d x$

Step 1.1.6

Factor $x$ out of $x \cdot (1 + 2 x) + x (3 x^{2})$ .

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

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Divide $1 + 2 x + 3 x^{2}$ by $1$ .

$\int 1 + 2 x + 3 x^{2} d x$

Step 2

Split the single integral into multiple integrals.

$\int d x + \int 2 x d x + \int 3 x^{2} d x$

Step 3

Apply the constant rule.

$x + C + \int 2 x d x + \int 3 x^{2} d x$

Step 4

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$x + C + 2 \int x d x + \int 3 x^{2} d x$

Step 5

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

$x + C + 2 (\frac{1}{2} x^{2} + C) + \int 3 x^{2} d x$

Step 6

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

$x + C + 2 (\frac{1}{2} x^{2} + C) + 3 \int x^{2} d x$

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Simplify.

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Step 8.2.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$x + x^{2} + 3 \frac{x^{3}}{3} + C$

Step 8.2.2

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

$x + x^{2} + \frac{3 x^{3}}{3} + C$

Step 8.2.3

Cancel the common factor of $3$ .

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Step 8.2.3.1

Cancel the common factor.

$x + x^{2} + \frac{3 x^{3}}{3} + C$

Step 8.2.3.2

Divide $x^{3}$ by $1$ .

$x + x^{2} + x^{3} + C$