Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $1 + x^{2}$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

$0 + \frac{d}{d x} [x^{2}]$

Step 1.1.4

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

$0 + 2 x$

Step 1.1.5

Add $0$ and $2 x$ .

$2 x$

Step 1.2

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

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

Step 1.3

Simplify.

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Step 1.3.1

One to any power is one.

$u_{lower} = 1 + 1$

Step 1.3.2

Add $1$ and $1$ .

$u_{lower} = 2$

Step 1.4

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

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

Step 1.5

Simplify.

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Step 1.5.1

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

$u_{upper} = 1 + 4$

Step 1.5.2

Add $1$ and $4$ .

$u_{upper} = 5$

Step 1.6

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

$u_{lower} = 2$

$u_{upper} = 5$

Step 1.7

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

$\int_{2}^{5} u d u$

Step 2

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

$\frac{1}{2} u^{2}]_{2}^{5}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $\frac{1}{2} u^{2}$ at $5$ and at $2$ .

$(\frac{1}{2} \cdot 5^{2}) - \frac{1}{2} \cdot 2^{2}$

Step 3.2

Simplify.

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Step 3.2.1

Raise $5$ to the power of $2$ .

$\frac{1}{2} \cdot 25 - \frac{1}{2} \cdot 2^{2}$

Step 3.2.2

Combine $\frac{1}{2}$ and $25$ .

$\frac{25}{2} - \frac{1}{2} \cdot 2^{2}$

Step 3.2.3

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

$\frac{25}{2} - \frac{1}{2} \cdot 4$

Step 3.2.4

Multiply $4$ by $- 1$ .

$\frac{25}{2} - 4 (\frac{1}{2})$

Step 3.2.5

Combine $- 4$ and $\frac{1}{2}$ .

$\frac{25}{2} + \frac{- 4}{2}$

Step 3.2.6

Cancel the common factor of $- 4$ and $2$ .

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Step 3.2.6.1

Factor $2$ out of $- 4$ .

$\frac{25}{2} + \frac{2 \cdot - 2}{2}$

Step 3.2.6.2

Cancel the common factors.

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Step 3.2.6.2.1

Factor $2$ out of $2$ .

$\frac{25}{2} + \frac{2 \cdot - 2}{2 (1)}$

Step 3.2.6.2.2

Cancel the common factor.

$\frac{25}{2} + \frac{2 \cdot - 2}{2 \cdot 1}$

Step 3.2.6.2.3

Rewrite the expression.

$\frac{25}{2} + \frac{- 2}{1}$

Step 3.2.6.2.4

Divide $- 2$ by $1$ .

$\frac{25}{2} - 2$

Step 3.2.7

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{25}{2} - 2 \cdot \frac{2}{2}$

Step 3.2.8

Combine $- 2$ and $\frac{2}{2}$ .

$\frac{25}{2} + \frac{- 2 \cdot 2}{2}$

Step 3.2.9

Combine the numerators over the common denominator.

$\frac{25 - 2 \cdot 2}{2}$

Step 3.2.10

Simplify the numerator.

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Step 3.2.10.1

Multiply $- 2$ by $2$ .

$\frac{25 - 4}{2}$

Step 3.2.10.2

Subtract $4$ from $25$ .

$\frac{21}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{21}{2}$

Decimal Form:

$10.5$

Mixed Number Form:

$10 \frac{1}{2}$

Step 5