Calculus Examples

$\int_{0}^{5} 3 x^{3} d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int_{0}^{5} x^{3} d x$

Step 2

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

$3 (\frac{1}{4} x^{4}]_{0}^{5})$

Step 3

Simplify the answer.

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Step 3.1

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

$3 (\frac{x^{4}}{4}]_{0}^{5})$

Step 3.2

Substitute and simplify.

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Step 3.2.1

Evaluate $\frac{x^{4}}{4}$ at $5$ and at $0$ .

$3 ((\frac{5^{4}}{4}) - \frac{0^{4}}{4})$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Raise $5$ to the power of $4$ .

$3 (\frac{625}{4} - \frac{0^{4}}{4})$

Step 3.2.2.2

Raising $0$ to any positive power yields $0$ .

$3 (\frac{625}{4} - \frac{0}{4})$

Step 3.2.2.3

Cancel the common factor of $0$ and $4$ .

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Step 3.2.2.3.1

Factor $4$ out of $0$ .

$3 (\frac{625}{4} - \frac{4 (0)}{4})$

Step 3.2.2.3.2

Cancel the common factors.

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Step 3.2.2.3.2.1

Factor $4$ out of $4$ .

$3 (\frac{625}{4} - \frac{4 \cdot 0}{4 \cdot 1})$

Step 3.2.2.3.2.2

Cancel the common factor.

$3 (\frac{625}{4} - \frac{4 \cdot 0}{4 \cdot 1})$

Step 3.2.2.3.2.3

Rewrite the expression.

$3 (\frac{625}{4} - \frac{0}{1})$

Step 3.2.2.3.2.4

Divide $0$ by $1$ .

$3 (\frac{625}{4} - 0)$

Step 3.2.2.4

Multiply $- 1$ by $0$ .

$3 (\frac{625}{4} + 0)$

Step 3.2.2.5

Add $\frac{625}{4}$ and $0$ .

$3 (\frac{625}{4})$

Step 3.2.2.6

Combine $3$ and $\frac{625}{4}$ .

$\frac{3 \cdot 625}{4}$

Step 3.2.2.7

Multiply $3$ by $625$ .

$\frac{1875}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{1875}{4}$

Decimal Form:

$468.75$

Mixed Number Form:

$468 \frac{3}{4}$

Step 5