Calculus Examples

$\int x^{4} (5 - x^{2}) d x$

Step 1

Multiply $x^{4} (5 - x^{2})$ .

$\int x^{4} \cdot 5 + x^{4} (- x^{2}) d x$

Step 2

Simplify.

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Step 2.1

Move $5$ to the left of $x^{4}$ .

$\int 5 \cdot x^{4} + x^{4} (- x^{2}) d x$

Step 2.2

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 2.2.1

Move $x^{2}$ .

$\int 5 x^{4} + x^{2} x^{4} \cdot - 1 d x$

Step 2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int 5 x^{4} + x^{2 + 4} \cdot - 1 d x$

Step 2.2.3

Add $2$ and $4$ .

$\int 5 x^{4} + x^{6} \cdot - 1 d x$

Step 2.3

Move $- 1$ to the left of $x^{6}$ .

$\int 5 x^{4} - 1 \cdot x^{6} d x$

Step 2.4

Rewrite $- 1 x^{6}$ as $- x^{6}$ .

$\int 5 x^{4} - x^{6} d x$

Step 3

Split the single integral into multiple integrals.

$\int 5 x^{4} d x + \int - x^{6} d x$

Step 4

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

$5 \int x^{4} d x + \int - x^{6} d x$

Step 5

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

$5 (\frac{1}{5} x^{5} + C) + \int - x^{6} d x$

Step 6

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

$5 (\frac{1}{5} x^{5} + C) - \int x^{6} d x$

Step 7

By the Power Rule, the integral of $x^{6}$ with respect to $x$ is $\frac{1}{7} x^{7}$ .

$5 (\frac{1}{5} x^{5} + C) - (\frac{1}{7} x^{7} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$5 (\frac{1}{5}) x^{5} - \frac{1}{7} x^{7} + C$

Step 8.2

Simplify.

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Step 8.2.1

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

$\frac{5}{5} x^{5} - \frac{1}{7} x^{7} + C$

Step 8.2.2

Cancel the common factor of $5$ .

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Step 8.2.2.1

Cancel the common factor.

$\frac{5}{5} x^{5} - \frac{1}{7} x^{7} + C$

Step 8.2.2.2

Rewrite the expression.

$1 x^{5} - \frac{1}{7} x^{7} + C$

Step 8.2.3

Multiply $x^{5}$ by $1$ .

$x^{5} - \frac{1}{7} x^{7} + C$