Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Multiply $5$ by $4$ .

$\int 20 x \cdot x^{2} d x$

Step 1.2

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

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Step 1.2.1

Move $x^{2}$ .

$\int 20 (x^{2} x) d x$

Step 1.2.2

Multiply $x^{2}$ by $x$ .

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Step 1.2.2.1

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

$\int 20 (x^{2} x^{1}) d x$

Step 1.2.2.2

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

$\int 20 x^{2 + 1} d x$

Step 1.2.3

Add $2$ and $1$ .

$\int 20 x^{3} d x$

Step 2

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

$20 \int x^{3} d x$

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

Rewrite $20 (\frac{1}{4} x^{4} + C)$ as $20 (\frac{1}{4}) x^{4} + C$ .

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

Step 4.2

Simplify.

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Step 4.2.1

Combine $20$ and $\frac{1}{4}$ .

$\frac{20}{4} x^{4} + C$

Step 4.2.2

Cancel the common factor of $20$ and $4$ .

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Step 4.2.2.1

Factor $4$ out of $20$ .

$\frac{4 \cdot 5}{4} x^{4} + C$

Step 4.2.2.2

Cancel the common factors.

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Step 4.2.2.2.1

Factor $4$ out of $4$ .

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

Step 4.2.2.2.2

Cancel the common factor.

$\frac{4 \cdot 5}{4 \cdot 1} x^{4} + C$

Step 4.2.2.2.3

Rewrite the expression.

$\frac{5}{1} x^{4} + C$

Step 4.2.2.2.4

Divide $5$ by $1$ .

$5 x^{4} + C$