Calculus Examples

$\int_{0}^{1} x^{3} {(2 x^{4} + 3)}^{2} d x$

Step 1

Let $u = 2 x^{4} + 3$ . Then $d u = 8 x^{3} d x$ , so $\frac{1}{8} d u = x^{3} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 2 x^{4} + 3$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $2 x^{4} + 3$ .

$\frac{d}{d x} [2 x^{4} + 3]$

Step 1.1.2

By the Sum Rule, the derivative of $2 x^{4} + 3$ with respect to $x$ is $\frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [3]$

Step 1.1.3

Evaluate $\frac{d}{d x} [2 x^{4}]$ .

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Step 1.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{4}$ with respect to $x$ is $2 \frac{d}{d x} [x^{4}]$ .

$2 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [3]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$2 (4 x^{3}) + \frac{d}{d x} [3]$

Step 1.1.3.3

Multiply $4$ by $2$ .

$8 x^{3} + \frac{d}{d x} [3]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$8 x^{3} + 0$

Step 1.1.4.2

Add $8 x^{3}$ and $0$ .

$8 x^{3}$

Step 1.2

Substitute the lower limit in for $x$ in $u = 2 x^{4} + 3$ .

$u_{lower} = 2 \cdot 0^{4} + 3$

Step 1.3

Simplify.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

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

$u_{lower} = 2 \cdot 0 + 3$

Step 1.3.1.2

Multiply $2$ by $0$ .

$u_{lower} = 0 + 3$

Step 1.3.2

Add $0$ and $3$ .

$u_{lower} = 3$

Step 1.4

Substitute the upper limit in for $x$ in $u = 2 x^{4} + 3$ .

$u_{upper} = 2 \cdot 1^{4} + 3$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

One to any power is one.

$u_{upper} = 2 \cdot 1 + 3$

Step 1.5.1.2

Multiply $2$ by $1$ .

$u_{upper} = 2 + 3$

Step 1.5.2

Add $2$ and $3$ .

$u_{upper} = 5$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 3$

$u_{upper} = 5$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{3}^{5} u^{2} \frac{1}{8} d u$

Step 2

Combine $u^{2}$ and $\frac{1}{8}$ .

$\int_{3}^{5} \frac{u^{2}}{8} d u$

Step 3

Since $\frac{1}{8}$ is constant with respect to $u$ , move $\frac{1}{8}$ out of the integral.

$\frac{1}{8} \int_{3}^{5} u^{2} d u$

Step 4

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

$\frac{1}{8} \frac{1}{3} u^{3}]_{3}^{5}$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{1}{3} u^{3}$ at $5$ and at $3$ .

$\frac{1}{8} ((\frac{1}{3} \cdot 5^{3}) - \frac{1}{3} \cdot 3^{3})$

Step 5.2

Simplify.

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Step 5.2.1

Raise $5$ to the power of $3$ .

$\frac{1}{8} (\frac{1}{3} \cdot 125 - \frac{1}{3} \cdot 3^{3})$

Step 5.2.2

Combine $\frac{1}{3}$ and $125$ .

$\frac{1}{8} (\frac{125}{3} - \frac{1}{3} \cdot 3^{3})$

Step 5.2.3

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

$\frac{1}{8} (\frac{125}{3} - \frac{1}{3} \cdot 27)$

Step 5.2.4

Multiply $27$ by $- 1$ .

$\frac{1}{8} (\frac{125}{3} - 27 (\frac{1}{3}))$

Step 5.2.5

Combine $- 27$ and $\frac{1}{3}$ .

$\frac{1}{8} (\frac{125}{3} + \frac{- 27}{3})$

Step 5.2.6

Cancel the common factor of $- 27$ and $3$ .

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Step 5.2.6.1

Factor $3$ out of $- 27$ .

$\frac{1}{8} (\frac{125}{3} + \frac{3 \cdot - 9}{3})$

Step 5.2.6.2

Cancel the common factors.

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Step 5.2.6.2.1

Factor $3$ out of $3$ .

$\frac{1}{8} (\frac{125}{3} + \frac{3 \cdot - 9}{3 (1)})$

Step 5.2.6.2.2

Cancel the common factor.

$\frac{1}{8} (\frac{125}{3} + \frac{3 \cdot - 9}{3 \cdot 1})$

Step 5.2.6.2.3

Rewrite the expression.

$\frac{1}{8} (\frac{125}{3} + \frac{- 9}{1})$

Step 5.2.6.2.4

Divide $- 9$ by $1$ .

$\frac{1}{8} (\frac{125}{3} - 9)$

Step 5.2.7

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{8} (\frac{125}{3} - 9 \cdot \frac{3}{3})$

Step 5.2.8

Combine $- 9$ and $\frac{3}{3}$ .

$\frac{1}{8} (\frac{125}{3} + \frac{- 9 \cdot 3}{3})$

Step 5.2.9

Combine the numerators over the common denominator.

$\frac{1}{8} \cdot \frac{125 - 9 \cdot 3}{3}$

Step 5.2.10

Simplify the numerator.

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Step 5.2.10.1

Multiply $- 9$ by $3$ .

$\frac{1}{8} \cdot \frac{125 - 27}{3}$

Step 5.2.10.2

Subtract $27$ from $125$ .

$\frac{1}{8} \cdot \frac{98}{3}$

Step 5.2.11

Multiply $\frac{1}{8}$ by $\frac{98}{3}$ .

$\frac{98}{8 \cdot 3}$

Step 5.2.12

Multiply $8$ by $3$ .

$\frac{98}{24}$

Step 5.2.13

Cancel the common factor of $98$ and $24$ .

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Step 5.2.13.1

Factor $2$ out of $98$ .

$\frac{2 (49)}{24}$

Step 5.2.13.2

Cancel the common factors.

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Step 5.2.13.2.1

Factor $2$ out of $24$ .

$\frac{2 \cdot 49}{2 \cdot 12}$

Step 5.2.13.2.2

Cancel the common factor.

$\frac{2 \cdot 49}{2 \cdot 12}$

Step 5.2.13.2.3

Rewrite the expression.

$\frac{49}{12}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{49}{12}$

Decimal Form:

$4.08\overline{3}$

Mixed Number Form:

$4 \frac{1}{12}$

Step 7