Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Expand $(x^{3} - 2 x^{2}) (\frac{1}{x} - 5)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Apply the distributive property.

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

Step 1.1.3

Apply the distributive property.

$\int x^{3} \frac{1}{x} + x^{3} \cdot - 5 - 2 x^{2} \frac{1}{x} - 2 x^{2} \cdot - 5 d x$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Cancel the common factor of $x$ .

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Step 1.2.1.1.1

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

$\int x \cdot x^{2} \frac{1}{x} + x^{3} \cdot - 5 - 2 x^{2} \frac{1}{x} - 2 x^{2} \cdot - 5 d x$

Step 1.2.1.1.2

Cancel the common factor.

$\int x \cdot x^{2} \frac{1}{x} + x^{3} \cdot - 5 - 2 x^{2} \frac{1}{x} - 2 x^{2} \cdot - 5 d x$

Step 1.2.1.1.3

Rewrite the expression.

$\int x^{2} + x^{3} \cdot - 5 - 2 x^{2} \frac{1}{x} - 2 x^{2} \cdot - 5 d x$

Step 1.2.1.2

Move $- 5$ to the left of $x^{3}$ .

$\int x^{2} - 5 \cdot x^{3} - 2 x^{2} \frac{1}{x} - 2 x^{2} \cdot - 5 d x$

Step 1.2.1.3

Cancel the common factor of $x$ .

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Step 1.2.1.3.1

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

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

Step 1.2.1.3.2

Cancel the common factor.

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

Step 1.2.1.3.3

Rewrite the expression.

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

Step 1.2.1.4

Multiply $- 5$ by $- 2$ .

$\int x^{2} - 5 x^{3} - 2 x + 10 x^{2} d x$

Step 1.2.2

Add $x^{2}$ and $10 x^{2}$ .

$\int 11 x^{2} - 5 x^{3} - 2 x d x$

Step 2

Split the single integral into multiple integrals.

$\int 11 x^{2} d x + \int - 5 x^{3} d x + \int - 2 x d x$

Step 3

Since $11$ is constant with respect to $x$ , move $11$ out of the integral.

$11 \int x^{2} d x + \int - 5 x^{3} d x + \int - 2 x d x$

Step 4

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

$11 (\frac{1}{3} x^{3} + C) + \int - 5 x^{3} d x + \int - 2 x d x$

Step 5

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

$11 (\frac{1}{3} x^{3} + C) - 5 \int x^{3} d x + \int - 2 x d x$

Step 6

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

$11 (\frac{1}{3} x^{3} + C) - 5 (\frac{1}{4} x^{4} + C) + \int - 2 x d x$

Step 7

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

$11 (\frac{1}{3} x^{3} + C) - 5 (\frac{1}{4} x^{4} + C) - 2 \int x d x$

Step 8

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

$11 (\frac{1}{3} x^{3} + C) - 5 (\frac{1}{4} x^{4} + C) - 2 (\frac{1}{2} x^{2} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

$\frac{11 x^{3}}{3} - \frac{5 x^{4}}{4} - 2 (\frac{1}{2} x^{2}) + C$

Step 9.2

Simplify.

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Step 9.2.1

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

$\frac{11 x^{3}}{3} - \frac{5 x^{4}}{4} - 2 \frac{x^{2}}{2} + C$

Step 9.2.2

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

$\frac{11 x^{3}}{3} - \frac{5 x^{4}}{4} + \frac{- 2 x^{2}}{2} + C$

Step 9.2.3

Cancel the common factor of $- 2$ and $2$ .

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Step 9.2.3.1

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

$\frac{11 x^{3}}{3} - \frac{5 x^{4}}{4} + \frac{2 (- x^{2})}{2} + C$

Step 9.2.3.2

Cancel the common factors.

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Step 9.2.3.2.1

Factor $2$ out of $2$ .

$\frac{11 x^{3}}{3} - \frac{5 x^{4}}{4} + \frac{2 (- x^{2})}{2 (1)} + C$

Step 9.2.3.2.2

Cancel the common factor.

$\frac{11 x^{3}}{3} - \frac{5 x^{4}}{4} + \frac{2 (- x^{2})}{2 \cdot 1} + C$

Step 9.2.3.2.3

Rewrite the expression.

$\frac{11 x^{3}}{3} - \frac{5 x^{4}}{4} + \frac{- x^{2}}{1} + C$

Step 9.2.3.2.4

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

$\frac{11 x^{3}}{3} - \frac{5 x^{4}}{4} - x^{2} + C$

Step 10

Reorder terms.

$\frac{11}{3} x^{3} - \frac{5}{4} x^{4} - x^{2} + C$