Calculus Examples

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

Step 1

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

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

Step 2

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$4 (\frac{1}{3} x^{3}]_{1}^{4})$

Step 3

Simplify the answer.

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Step 3.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$4 (\frac{x^{3}}{3}]_{1}^{4})$

Step 3.2

Substitute and simplify.

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Step 3.2.1

Evaluate $\frac{x^{3}}{3}$ at $4$ and at $1$ .

$4 ((\frac{4^{3}}{3}) - \frac{1^{3}}{3})$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Raise $4$ to the power of $3$ .

$4 (\frac{64}{3} - \frac{1^{3}}{3})$

Step 3.2.2.2

One to any power is one.

$4 (\frac{64}{3} - \frac{1}{3})$

Step 3.2.2.3

Combine the numerators over the common denominator.

$4 \frac{64 - 1}{3}$

Step 3.2.2.4

Subtract $1$ from $64$ .

$4 (\frac{63}{3})$

Step 3.2.2.5

Cancel the common factor of $63$ and $3$ .

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Step 3.2.2.5.1

Factor $3$ out of $63$ .

$4 \frac{3 \cdot 21}{3}$

Step 3.2.2.5.2

Cancel the common factors.

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Step 3.2.2.5.2.1

Factor $3$ out of $3$ .

$4 \frac{3 \cdot 21}{3 (1)}$

Step 3.2.2.5.2.2

Cancel the common factor.

$4 \frac{3 \cdot 21}{3 \cdot 1}$

Step 3.2.2.5.2.3

Rewrite the expression.

$4 (\frac{21}{1})$

Step 3.2.2.5.2.4

Divide $21$ by $1$ .

$4 \cdot 21$

Step 3.2.2.6

Multiply $4$ by $21$ .

$84$

Step 4