Calculus Examples

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

Step 1

Split the fraction into multiple fractions.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Factor $x$ out of $5 x$ .

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

Step 1.1.3

Factor $x$ out of $x (2 x) + x \cdot 5$ .

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

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Divide $2 x + 5$ by $1$ .

$\int 2 x + 5 d x$

Step 2

Split the single integral into multiple integrals.

$\int 2 x d x + \int 5 d x$

Step 3

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

$2 \int x d x + \int 5 d x$

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

Simplify.

$x^{2} + 5 x + C$