Calculus Examples

$\int (\frac{x^{4} - 6 x^{3} - 7 x}{x}) d x$

Step 1

Remove parentheses.

$\int \frac{x^{4} - 6 x^{3} - 7 x}{x} d x$

Step 2

Split the fraction into multiple fractions.

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Step 2.1

Factor $x$ out of $x^{4} - 6 x^{3} - 7 x$ .

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Step 2.1.1

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

$\int \frac{x \cdot x^{3} - 6 x^{3} - 7 x}{x} d x$

Step 2.1.2

Factor $x$ out of $- 6 x^{3}$ .

$\int \frac{x \cdot x^{3} + x (- 6 x^{2}) - 7 x}{x} d x$

Step 2.1.3

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

$\int \frac{x \cdot x^{3} + x (- 6 x^{2}) + x \cdot - 7}{x} d x$

Step 2.1.4

Factor $x$ out of $x \cdot x^{3} + x (- 6 x^{2})$ .

$\int \frac{x (x^{3} - 6 x^{2}) + x \cdot - 7}{x} d x$

Step 2.1.5

Factor $x$ out of $x (x^{3} - 6 x^{2}) + x \cdot - 7$ .

$\int \frac{x (x^{3} - 6 x^{2} - 7)}{x} d x$

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

$\int \frac{x (x^{3} - 6 x^{2} - 7)}{x} d x$

Step 2.2.2

Divide $x^{3} - 6 x^{2} - 7$ by $1$ .

$\int x^{3} - 6 x^{2} - 7 d x$

Step 3

Split the single integral into multiple integrals.

$\int x^{3} d x + \int - 6 x^{2} d x + \int - 7 d x$

Step 4

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

$\frac{1}{4} x^{4} + C + \int - 6 x^{2} d x + \int - 7 d x$

Step 5

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

$\frac{1}{4} x^{4} + C - 6 \int x^{2} d x + \int - 7 d x$

Step 6

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

$\frac{1}{4} x^{4} + C - 6 (\frac{1}{3} x^{3} + C) + \int - 7 d x$

Step 7

Apply the constant rule.

$\frac{1}{4} x^{4} + C - 6 (\frac{1}{3} x^{3} + C) - 7 x + C$

Step 8

Simplify.

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Step 8.1

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

$\frac{1}{4} x^{4} + C - 6 (\frac{x^{3}}{3} + C) - 7 x + C$

Step 8.2

Simplify.

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

Step 8.3

Reorder terms.

$\frac{1}{4} x^{4} - 2 x^{3} - 7 x + C$