Calculus Examples

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

Step 1

Let $u = 2 x^{4} + 1$ . 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 $u = 2 x^{4} + 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $2 x^{4} + 1$ .

$\frac{d}{d x} [2 x^{4} + 1]$

Step 1.1.2

By the Sum Rule, the derivative of $2 x^{4} + 1$ with respect to $x$ is $\frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [1]$

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

$2 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [1]$

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 $n = 4$ .

$2 (4 x^{3}) + \frac{d}{d x} [1]$

Step 1.1.3.3

Multiply $4$ by $2$ .

$8 x^{3} + \frac{d}{d x} [1]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ 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 $u = 2 x^{4} + 1$ .

$u_{lower} = 2 \cdot 0^{4} + 1$

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 $0$ .

$u_{lower} = 2 \cdot 0 + 1$

Step 1.3.1.2

Multiply $2$ by $0$ .

$u_{lower} = 0 + 1$

Step 1.3.2

Add $0$ and $1$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = 2 x^{4} + 1$ .

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

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.

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

Step 1.5.1.2

Multiply $2$ by $1$ .

$u_{upper} = 2 + 1$

Step 1.5.2

Add $2$ and $1$ .

$u_{upper} = 3$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 3$

Step 1.7

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify the expression.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Simplify.

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Step 5.3.1

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

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

Step 5.3.2

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

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Step 5.3.2.1

Factor $3$ out of $27$ .

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

Step 5.3.2.2

Cancel the common factors.

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Step 5.3.2.2.1

Factor $3$ out of $3$ .

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

Step 5.3.2.2.2

Cancel the common factor.

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

Step 5.3.2.2.3

Rewrite the expression.

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

Step 5.3.2.2.4

Divide $9$ by $1$ .

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

Step 5.4

Simplify the expression.

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Step 5.4.1

One to any power is one.

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

Step 5.4.2

Multiply $- 1$ by $1$ .

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

Step 5.5

Simplify.

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Step 5.5.1

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

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

Step 5.5.2

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

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

Step 5.5.3

Combine the numerators over the common denominator.

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

Step 5.5.4

Simplify the numerator.

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Step 5.5.4.1

Multiply $9$ by $3$ .

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

Step 5.5.4.2

Subtract $1$ from $27$ .

$\frac{1}{8} \cdot \frac{26}{3}$

Step 5.6

Simplify.

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Step 5.6.1

Multiply $\frac{1}{8}$ by $\frac{26}{3}$ .

$\frac{26}{8 \cdot 3}$

Step 5.6.2

Multiply $8$ by $3$ .

$\frac{26}{24}$

Step 5.6.3

Cancel the common factor of $26$ and $24$ .

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Step 5.6.3.1

Factor $2$ out of $26$ .

$\frac{2 (13)}{24}$

Step 5.6.3.2

Cancel the common factors.

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Step 5.6.3.2.1

Factor $2$ out of $24$ .

$\frac{2 \cdot 13}{2 \cdot 12}$

Step 5.6.3.2.2

Cancel the common factor.

$\frac{2 \cdot 13}{2 \cdot 12}$

Step 5.6.3.2.3

Rewrite the expression.

$\frac{13}{12}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{13}{12}$

Decimal Form:

$1.08\overline{3}$

Mixed Number Form:

$1 \frac{1}{12}$