Calculus Examples

$\int \frac{12 x^{4} + 14 x^{3} + x^{2}}{x^{2}} d x$

Step 1

Simplify.

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Step 1.1

Factor $x^{2}$ out of $12 x^{4} + 14 x^{3} + x^{2}$ .

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Step 1.1.1

Factor $x^{2}$ out of $12 x^{4}$ .

$\int \frac{x^{2} (12 x^{2}) + 14 x^{3} + x^{2}}{x^{2}} d x$

Step 1.1.2

Factor $x^{2}$ out of $14 x^{3}$ .

$\int \frac{x^{2} (12 x^{2}) + x^{2} (14 x) + x^{2}}{x^{2}} d x$

Step 1.1.3

Multiply by $1$ .

$\int \frac{x^{2} (12 x^{2}) + x^{2} (14 x) + x^{2} \cdot 1}{x^{2}} d x$

Step 1.1.4

Factor $x^{2}$ out of $x^{2} (12 x^{2}) + x^{2} (14 x)$ .

$\int \frac{x^{2} (12 x^{2} + 14 x) + x^{2} \cdot 1}{x^{2}} d x$

Step 1.1.5

Factor $x^{2}$ out of $x^{2} (12 x^{2} + 14 x) + x^{2} \cdot 1$ .

$\int \frac{x^{2} (12 x^{2} + 14 x + 1)}{x^{2}} d x$

Step 1.2

Cancel the common factor of $x^{2}$ .

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Step 1.2.1

Cancel the common factor.

$\int \frac{x^{2} (12 x^{2} + 14 x + 1)}{x^{2}} d x$

Step 1.2.2

Divide $12 x^{2} + 14 x + 1$ by $1$ .

$\int 12 x^{2} + 14 x + 1 d x$

Step 2

Split the single integral into multiple integrals.

$\int 12 x^{2} d x + \int 14 x d x + \int d x$

Step 3

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$12 \int x^{2} d x + \int 14 x d x + \int d x$

Step 4

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

$12 (\frac{1}{3} x^{3} + C) + \int 14 x d x + \int d x$

Step 5

Since $14$ is constant with respect to $x$ , move $14$ out of the integral.

$12 (\frac{1}{3} x^{3} + C) + 14 \int x d x + \int d x$

Step 6

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

$12 (\frac{1}{3} x^{3} + C) + 14 (\frac{1}{2} x^{2} + C) + \int d x$

Step 7

Apply the constant rule.

$12 (\frac{1}{3} x^{3} + C) + 14 (\frac{1}{2} x^{2} + C) + x + C$

Step 8

Simplify.

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Step 8.1

Simplify.

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Step 8.1.1

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

$12 (\frac{x^{3}}{3} + C) + 14 (\frac{1}{2} x^{2} + C) + x + C$

Step 8.1.2

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

$12 (\frac{x^{3}}{3} + C) + 14 (\frac{x^{2}}{2} + C) + x + C$

Step 8.2

Simplify.

$4 x^{3} + 7 x^{2} + x + C$