Calculus Examples

$\int \frac{8 x^{4} + 18 x^{3} - 7 x}{x} d x$

Step 1

Split the fraction into multiple fractions.

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Step 1.1

Factor $x$ out of $8 x^{4} + 18 x^{3} - 7 x$ .

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Step 1.1.1

Factor $x$ out of $8 x^{4}$ .

$\int \frac{x (8 x^{3}) + 18 x^{3} - 7 x}{x} d x$

Step 1.1.2

Factor $x$ out of $18 x^{3}$ .

$\int \frac{x (8 x^{3}) + x (18 x^{2}) - 7 x}{x} d x$

Step 1.1.3

Factor $x$ out of $- 7 x$ .

$\int \frac{x (8 x^{3}) + x (18 x^{2}) + x \cdot - 7}{x} d x$

Step 1.1.4

Factor $x$ out of $x (8 x^{3}) + x (18 x^{2})$ .

$\int \frac{x (8 x^{3} + 18 x^{2}) + x \cdot - 7}{x} d x$

Step 1.1.5

Factor $x$ out of $x (8 x^{3} + 18 x^{2}) + x \cdot - 7$ .

$\int \frac{x (8 x^{3} + 18 x^{2} - 7)}{x} d x$

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

$\int \frac{x (8 x^{3} + 18 x^{2} - 7)}{x} d x$

Step 1.2.2

Divide $8 x^{3} + 18 x^{2} - 7$ by $1$ .

$\int 8 x^{3} + 18 x^{2} - 7 d x$

Step 2

Split the single integral into multiple integrals.

$\int 8 x^{3} d x + \int 18 x^{2} d x + \int - 7 d x$

Step 3

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$8 \int x^{3} d x + \int 18 x^{2} d x + \int - 7 d x$

Step 4

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$8 (\frac{1}{4} x^{4} + C) + \int 18 x^{2} d x + \int - 7 d x$

Step 5

Since $18$ is constant with respect to $x$ , move $18$ out of the integral.

$8 (\frac{1}{4} x^{4} + C) + 18 \int x^{2} d x + \int - 7 d x$

Step 6

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

$8 (\frac{1}{4} x^{4} + C) + 18 (\frac{1}{3} x^{3} + C) + \int - 7 d x$

Step 7

Apply the constant rule.

$8 (\frac{1}{4} x^{4} + C) + 18 (\frac{1}{3} x^{3} + C) - 7 x + C$

Step 8

Simplify.

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Step 8.1

Simplify.

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Step 8.1.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$8 (\frac{x^{4}}{4} + C) + 18 (\frac{1}{3} x^{3} + C) - 7 x + C$

Step 8.1.2

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

$8 (\frac{x^{4}}{4} + C) + 18 (\frac{x^{3}}{3} + C) - 7 x + C$

Step 8.2

Simplify.

$2 x^{4} + 6 x^{3} - 7 x + C$