Calculus Examples

$\int (\frac{x^{6} - 5 x}{x^{2}}) d x$

Step 1

Remove parentheses.

$\int \frac{x^{6} - 5 x}{x^{2}} d x$

Step 2

Simplify.

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Step 2.1

Factor $x$ out of $x^{6} - 5 x$ .

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Step 2.1.1

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

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

Step 2.1.2

Factor $x$ out of $- 5 x$ .

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

Step 2.1.3

Factor $x$ out of $x \cdot x^{5} + x \cdot - 5$ .

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

Step 2.2

Cancel the common factors.

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Step 2.2.1

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

$\int \frac{x (x^{5} - 5)}{x \cdot x} d x$

Step 2.2.2

Cancel the common factor.

$\int \frac{x (x^{5} - 5)}{x \cdot x} d x$

Step 2.2.3

Rewrite the expression.

$\int \frac{x^{5} - 5}{x} d x$

Step 3

Divide $x^{5} - 5$ by $x$ .

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Step 3.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$0$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$5$

Step 3.2

Divide the highest order term in the dividend $x^{5}$ by the highest order term in divisor $x$ .

$x^{4}$

$x$

$0$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$5$

Step 3.3

Multiply the new quotient term by the divisor.

$x^{4}$

$x$

$0$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$5$

$x^{5}$

$0$

Step 3.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{5} + 0$

$x^{4}$

$x$

$0$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$5$

$x^{5}$

$0$

Step 3.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$x^{4}$

$x$

$0$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$5$

$x^{5}$

$0$

Step 3.6

Pull the next term from the original dividend down into the current dividend.

$x^{4}$

$x$

$0$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$5$

$x^{5}$

$0$

$0 x^{3}$

$0 x^{2}$

Step 3.7

The final answer is the quotient plus the remainder over the divisor.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

$\frac{1}{5} x^{5} + C + \int - \frac{5}{x} d x$

Step 6

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{5} x^{5} + C - \int \frac{5}{x} d x$

Step 7

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

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

Step 8

Multiply $5$ by $- 1$ .

$\frac{1}{5} x^{5} + C - 5 \int \frac{1}{x} d x$

Step 9

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$\frac{1}{5} x^{5} + C - 5 (\ln (| x |) + C)$

Step 10

Simplify.

$\frac{1}{5} x^{5} - 5 \ln (| x |) + C$