Calculus Examples

$\int \frac{20 x^{5} - 20 x^{4} + x}{x} d x$

Step 1

Split the fraction into multiple fractions.

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Step 1.1

Factor $x$ out of $20 x^{5} - 20 x^{4} + x$ .

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Step 1.1.1

Factor $x$ out of $20 x^{5}$ .

$\int \frac{x (20 x^{4}) - 20 x^{4} + x}{x} d x$

Step 1.1.2

Factor $x$ out of $- 20 x^{4}$ .

$\int \frac{x (20 x^{4}) + x (- 20 x^{3}) + x}{x} d x$

Step 1.1.3

Raise $x$ to the power of $1$ .

$\int \frac{x (20 x^{4}) + x (- 20 x^{3}) + x^{1}}{x} d x$

Step 1.1.4

Factor $x$ out of $x^{1}$ .

$\int \frac{x (20 x^{4}) + x (- 20 x^{3}) + x \cdot 1}{x} d x$

Step 1.1.5

Factor $x$ out of $x (20 x^{4}) + x (- 20 x^{3})$ .

$\int \frac{x (20 x^{4} - 20 x^{3}) + x \cdot 1}{x} d x$

Step 1.1.6

Factor $x$ out of $x (20 x^{4} - 20 x^{3}) + x \cdot 1$ .

$\int \frac{x (20 x^{4} - 20 x^{3} + 1)}{x} d x$

Step 1.2

Cancel the common factor of $x$ .

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Step 1.2.1

Cancel the common factor.

$\int \frac{x (20 x^{4} - 20 x^{3} + 1)}{x} d x$

Step 1.2.2

Divide $20 x^{4} - 20 x^{3} + 1$ by $1$ .

$\int 20 x^{4} - 20 x^{3} + 1 d x$

Step 2

Split the single integral into multiple integrals.

$\int 20 x^{4} d x + \int - 20 x^{3} d x + \int d x$

Step 3

Since $20$ is constant with respect to $x$ , move $20$ out of the integral.

$20 \int x^{4} d x + \int - 20 x^{3} d x + \int d x$

Step 4

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$20 (\frac{1}{5} x^{5} + C) + \int - 20 x^{3} d x + \int d x$

Step 5

Since $- 20$ is constant with respect to $x$ , move $- 20$ out of the integral.

$20 (\frac{1}{5} x^{5} + C) - 20 \int x^{3} d x + \int d x$

Step 6

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

$20 (\frac{1}{5} x^{5} + C) - 20 (\frac{1}{4} x^{4} + C) + \int d x$

Step 7

Apply the constant rule.

$20 (\frac{1}{5} x^{5} + C) - 20 (\frac{1}{4} x^{4} + C) + x + C$

Step 8

Simplify.

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Step 8.1

Simplify.

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Step 8.1.1

Combine $\frac{1}{5}$ and $x^{5}$ .

$20 (\frac{x^{5}}{5} + C) - 20 (\frac{1}{4} x^{4} + C) + x + C$

Step 8.1.2

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

$20 (\frac{x^{5}}{5} + C) - 20 (\frac{x^{4}}{4} + C) + x + C$

Step 8.2

Simplify.

$4 x^{5} - 5 x^{4} + x + C$