Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $x^{2} + 1$ .

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

Step 1.1.2

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

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

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 1.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$2 x + 0$

Step 1.1.5

Add $2 x$ and $0$ .

$2 x$

Step 1.2

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

$u_{lower} = 0^{2} + 1$

Step 1.3

Simplify.

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Step 1.3.1

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

$u_{lower} = 0 + 1$

Step 1.3.2

Add $0$ and $1$ .

$u_{lower} = 1$

Step 1.4

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

$u_{upper} = 1^{2} + 1$

Step 1.5

Simplify.

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Step 1.5.1

One to any power is one.

$u_{upper} = 1 + 1$

Step 1.5.2

Add $1$ and $1$ .

$u_{upper} = 2$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 2$

Step 1.7

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

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

Step 2

Combine $u^{3}$ and $\frac{1}{2}$ .

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

Step 3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 4

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

$\frac{1}{2} \frac{1}{4} u^{4}]_{1}^{2}$

Step 5

Simplify the expression.

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Step 5.1

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

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

Step 5.2

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

$\frac{1}{2} (\frac{1}{4} \cdot 16 - \frac{1}{4} \cdot 1^{4})$

Step 5.3

Simplify.

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Step 5.3.1

Combine $\frac{1}{4}$ and $16$ .

$\frac{1}{2} (\frac{16}{4} - \frac{1}{4} \cdot 1^{4})$

Step 5.3.2

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

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Step 5.3.2.1

Factor $4$ out of $16$ .

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

Step 5.3.2.2

Cancel the common factors.

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Step 5.3.2.2.1

Factor $4$ out of $4$ .

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

Step 5.3.2.2.2

Cancel the common factor.

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

Step 5.3.2.2.3

Rewrite the expression.

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

Step 5.3.2.2.4

Divide $4$ by $1$ .

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

Step 5.4

Simplify the expression.

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Step 5.4.1

One to any power is one.

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

Step 5.4.2

Multiply $- 1$ by $1$ .

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

Step 5.5

Simplify.

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Step 5.5.1

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

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

Step 5.5.2

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

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

Step 5.5.3

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \frac{4 \cdot 4 - 1}{4}$

Step 5.5.4

Simplify the numerator.

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Step 5.5.4.1

Multiply $4$ by $4$ .

$\frac{1}{2} \cdot \frac{16 - 1}{4}$

Step 5.5.4.2

Subtract $1$ from $16$ .

$\frac{1}{2} \cdot \frac{15}{4}$

Step 5.6

Simplify.

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Step 5.6.1

Multiply $\frac{1}{2}$ by $\frac{15}{4}$ .

$\frac{15}{2 \cdot 4}$

Step 5.6.2

Multiply $2$ by $4$ .

$\frac{15}{8}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{15}{8}$

Decimal Form:

$1.875$

Mixed Number Form:

$1 \frac{7}{8}$