Calculus Examples

$\int_{1}^{2} 4 x^{3} d x$

Step 1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int_{1}^{2} x^{3} d x$

Step 2

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

$4 (\frac{1}{4} x^{4}]_{1}^{2})$

Step 3

Simplify the answer.

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Step 3.1

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

$4 (\frac{x^{4}}{4}]_{1}^{2})$

Step 3.2

Substitute and simplify.

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Step 3.2.1

Evaluate $\frac{x^{4}}{4}$ at $2$ and at $1$ .

$4 ((\frac{2^{4}}{4}) - \frac{1^{4}}{4})$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Raise $2$ to the power of $4$ .

$4 (\frac{16}{4} - \frac{1^{4}}{4})$

Step 3.2.2.2

Cancel the common factor of $16$ and $4$ .

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Step 3.2.2.2.1

Factor $4$ out of $16$ .

$4 (\frac{4 \cdot 4}{4} - \frac{1^{4}}{4})$

Step 3.2.2.2.2

Cancel the common factors.

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Step 3.2.2.2.2.1

Factor $4$ out of $4$ .

$4 (\frac{4 \cdot 4}{4 (1)} - \frac{1^{4}}{4})$

Step 3.2.2.2.2.2

Cancel the common factor.

$4 (\frac{4 \cdot 4}{4 \cdot 1} - \frac{1^{4}}{4})$

Step 3.2.2.2.2.3

Rewrite the expression.

$4 (\frac{4}{1} - \frac{1^{4}}{4})$

Step 3.2.2.2.2.4

Divide $4$ by $1$ .

$4 (4 - \frac{1^{4}}{4})$

Step 3.2.2.3

One to any power is one.

$4 (4 - \frac{1}{4})$

Step 3.2.2.4

To write $4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$4 (4 \cdot \frac{4}{4} - \frac{1}{4})$

Step 3.2.2.5

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

$4 (\frac{4 \cdot 4}{4} - \frac{1}{4})$

Step 3.2.2.6

Combine the numerators over the common denominator.

$4 \frac{4 \cdot 4 - 1}{4}$

Step 3.2.2.7

Simplify the numerator.

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Step 3.2.2.7.1

Multiply $4$ by $4$ .

$4 \frac{16 - 1}{4}$

Step 3.2.2.7.2

Subtract $1$ from $16$ .

$4 (\frac{15}{4})$

Step 3.2.2.8

Combine $4$ and $\frac{15}{4}$ .

$\frac{4 \cdot 15}{4}$

Step 3.2.2.9

Multiply $4$ by $15$ .

$\frac{60}{4}$

Step 3.2.2.10

Cancel the common factor of $60$ and $4$ .

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Step 3.2.2.10.1

Factor $4$ out of $60$ .

$\frac{4 \cdot 15}{4}$

Step 3.2.2.10.2

Cancel the common factors.

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Step 3.2.2.10.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 15}{4 (1)}$

Step 3.2.2.10.2.2

Cancel the common factor.

$\frac{4 \cdot 15}{4 \cdot 1}$

Step 3.2.2.10.2.3

Rewrite the expression.

$\frac{15}{1}$

Step 3.2.2.10.2.4

Divide $15$ by $1$ .

$15$

Step 4