Calculus Examples

$\int (\frac{2 x^{3} + 5 x^{2} - 2 x}{x}) d x$

Step 1

Remove parentheses.

$\int \frac{2 x^{3} + 5 x^{2} - 2 x}{x} d x$

Step 2

Split the fraction into multiple fractions.

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Step 2.1

Factor $x$ out of $2 x^{3} + 5 x^{2} - 2 x$ .

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Step 2.1.1

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

$\int \frac{x (2 x^{2}) + 5 x^{2} - 2 x}{x} d x$

Step 2.1.2

Factor $x$ out of $5 x^{2}$ .

$\int \frac{x (2 x^{2}) + x (5 x) - 2 x}{x} d x$

Step 2.1.3

Factor $x$ out of $- 2 x$ .

$\int \frac{x (2 x^{2}) + x (5 x) + x \cdot - 2}{x} d x$

Step 2.1.4

Factor $x$ out of $x (2 x^{2}) + x (5 x)$ .

$\int \frac{x (2 x^{2} + 5 x) + x \cdot - 2}{x} d x$

Step 2.1.5

Factor $x$ out of $x (2 x^{2} + 5 x) + x \cdot - 2$ .

$\int \frac{x (2 x^{2} + 5 x - 2)}{x} d x$

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

$\int \frac{x (2 x^{2} + 5 x - 2)}{x} d x$

Step 2.2.2

Divide $2 x^{2} + 5 x - 2$ by $1$ .

$\int 2 x^{2} + 5 x - 2 d x$

Step 3

Split the single integral into multiple integrals.

$\int 2 x^{2} d x + \int 5 x d x + \int - 2 d x$

Step 4

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

$2 \int x^{2} d x + \int 5 x d x + \int - 2 d x$

Step 5

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

$2 (\frac{1}{3} x^{3} + C) + \int 5 x d x + \int - 2 d x$

Step 6

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$2 (\frac{1}{3} x^{3} + C) + 5 \int x d x + \int - 2 d x$

Step 7

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

$2 (\frac{1}{3} x^{3} + C) + 5 (\frac{1}{2} x^{2} + C) + \int - 2 d x$

Step 8

Apply the constant rule.

$2 (\frac{1}{3} x^{3} + C) + 5 (\frac{1}{2} x^{2} + C) - 2 x + C$

Step 9

Simplify.

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Step 9.1

Simplify.

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Step 9.1.1

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

$2 (\frac{x^{3}}{3} + C) + 5 (\frac{1}{2} x^{2} + C) - 2 x + C$

Step 9.1.2

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

$2 (\frac{x^{3}}{3} + C) + 5 (\frac{x^{2}}{2} + C) - 2 x + C$

Step 9.2

Simplify.

$\frac{2 x^{3}}{3} + \frac{5 x^{2}}{2} - 2 x + C$

Step 9.3

Reorder terms.

$\frac{2}{3} x^{3} + \frac{5}{2} x^{2} - 2 x + C$