Calculus Examples

$\int_{1}^{e} \frac{x^{2} + 1}{x} d x$

Step 1

Split the fraction into multiple fractions.

$\int_{1}^{e} \frac{x^{2}}{x} + \frac{1}{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int_{1}^{e} \frac{x^{2}}{x} d x + \int_{1}^{e} \frac{1}{x} d x$

Step 3

Cancel the common factor of $x^{2}$ and $x$ .

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Step 3.1

Factor $x$ out of $x^{2}$ .

$\int_{1}^{e} \frac{x \cdot x}{x} d x + \int_{1}^{e} \frac{1}{x} d x$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Raise $x$ to the power of $1$ .

$\int_{1}^{e} \frac{x \cdot x}{x^{1}} d x + \int_{1}^{e} \frac{1}{x} d x$

Step 3.2.2

Factor $x$ out of $x^{1}$ .

$\int_{1}^{e} \frac{x \cdot x}{x \cdot 1} d x + \int_{1}^{e} \frac{1}{x} d x$

Step 3.2.3

Cancel the common factor.

$\int_{1}^{e} \frac{x \cdot x}{x \cdot 1} d x + \int_{1}^{e} \frac{1}{x} d x$

Step 3.2.4

Rewrite the expression.

$\int_{1}^{e} \frac{x}{1} d x + \int_{1}^{e} \frac{1}{x} d x$

Step 3.2.5

Divide $x$ by $1$ .

$\int_{1}^{e} x d x + \int_{1}^{e} \frac{1}{x} d x$

Step 4

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

$\frac{1}{2} x^{2}]_{1}^{e} + \int_{1}^{e} \frac{1}{x} d x$

Step 5

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$\frac{1}{2} x^{2}]_{1}^{e} + \ln (| x |)]_{1}^{e}$

Step 6

Simplify the answer.

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Step 6.1

Combine $\frac{1}{2} x^{2}]_{1}^{e}$ and $\ln (| x |)]_{1}^{e}$ .

$\frac{1}{2} x^{2} + \ln (| x |)]_{1}^{e}$

Step 6.2

Substitute and simplify.

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Step 6.2.1

Evaluate $\frac{1}{2} x^{2} + \ln (| x |)$ at $e$ and at $1$ .

$(\frac{1}{2} e^{2} + \ln (| e |)) - (\frac{1}{2} \cdot 1^{2} + \ln (| 1 |))$

Step 6.2.2

Simplify.

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Step 6.2.2.1

Combine $\frac{1}{2}$ and $e^{2}$ .

$\frac{e^{2}}{2} + \ln (| e |) - (\frac{1}{2} \cdot 1^{2} + \ln (| 1 |))$

Step 6.2.2.2

One to any power is one.

$\frac{e^{2}}{2} + \ln (| e |) - (\frac{1}{2} \cdot 1 + \ln (| 1 |))$

Step 6.2.2.3

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

$\frac{e^{2}}{2} + \ln (| e |) - (\frac{1}{2} + \ln (| 1 |))$

Step 6.2.2.4

To write $- (\frac{1}{2} + \ln (| 1 |))$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{e^{2}}{2} - (\frac{1}{2} + \ln (| 1 |)) \cdot \frac{2}{2} + \ln (| e |)$

Step 6.2.2.5

Combine $- (\frac{1}{2} + \ln (| 1 |))$ and $\frac{2}{2}$ .

$\frac{e^{2}}{2} + \frac{- (\frac{1}{2} + \ln (| 1 |)) \cdot 2}{2} + \ln (| e |)$

Step 6.2.2.6

Combine the numerators over the common denominator.

$\frac{e^{2} - (\frac{1}{2} + \ln (| 1 |)) \cdot 2}{2} + \ln (| e |)$

Step 6.2.2.7

Multiply $2$ by $- 1$ .

$\frac{e^{2} - 2 (\frac{1}{2} + \ln (| 1 |))}{2} + \ln (| e |)$

Step 7

Simplify.

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Step 7.1

Simplify each term.

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Step 7.1.1

Simplify the numerator.

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Step 7.1.1.1

Simplify each term.

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Step 7.1.1.1.1

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{e^{2} - 2 (\frac{1}{2} + \ln (1))}{2} + \ln (| e |)$

Step 7.1.1.1.2

The natural logarithm of $1$ is $0$ .

$\frac{e^{2} - 2 (\frac{1}{2} + 0)}{2} + \ln (| e |)$

Step 7.1.1.2

Add $\frac{1}{2}$ and $0$ .

$\frac{e^{2} - 2 (\frac{1}{2})}{2} + \ln (| e |)$

Step 7.1.1.3

Cancel the common factor of $2$ .

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Step 7.1.1.3.1

Factor $2$ out of $- 2$ .

$\frac{e^{2} + 2 (- 1) \frac{1}{2}}{2} + \ln (| e |)$

Step 7.1.1.3.2

Cancel the common factor.

$\frac{e^{2} + 2 \cdot - 1 \frac{1}{2}}{2} + \ln (| e |)$

Step 7.1.1.3.3

Rewrite the expression.

$\frac{e^{2} - 1}{2} + \ln (| e |)$

Step 7.1.1.4

Rewrite $e^{2} - 1$ in a factored form.

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Step 7.1.1.4.1

Rewrite $1$ as $1^{2}$ .

$\frac{e^{2} - 1^{2}}{2} + \ln (| e |)$

Step 7.1.1.4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = e$ and $b = 1$ .

$\frac{(e + 1) (e - 1)}{2} + \ln (| e |)$

Step 7.1.2

$e$ is approximately $2.71828182$ which is positive so remove the absolute value

$\frac{(e + 1) (e - 1)}{2} + \ln (e)$

Step 7.1.3

The natural logarithm of $e$ is $1$ .

$\frac{(e + 1) (e - 1)}{2} + 1$

Step 7.2

Write $1$ as a fraction with a common denominator.

$\frac{(e + 1) (e - 1)}{2} + \frac{2}{2}$

Step 7.3

Combine the numerators over the common denominator.

$\frac{(e + 1) (e - 1) + 2}{2}$

Step 7.4

Simplify the numerator.

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Step 7.4.1

Expand $(e + 1) (e - 1)$ using the FOIL Method.

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Step 7.4.1.1

Apply the distributive property.

$\frac{e (e - 1) + 1 (e - 1) + 2}{2}$

Step 7.4.1.2

Apply the distributive property.

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

Step 7.4.1.3

Apply the distributive property.

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

Step 7.4.2

Simplify and combine like terms.

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Step 7.4.2.1

Simplify each term.

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Step 7.4.2.1.1

Move $- 1$ to the left of $e$ .

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

Step 7.4.2.1.2

Rewrite $- 1 e$ as $- e$ .

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

Step 7.4.2.1.3

Multiply $e$ by $1$ .

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

Step 7.4.2.1.4

Multiply $- 1$ by $1$ .

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

Step 7.4.2.2

Add $- e$ and $e$ .

$\frac{e e + 0 - 1 + 2}{2}$

Step 7.4.2.3

Add $e e$ and $0$ .

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

Step 7.4.3

Add $- 1$ and $2$ .

$\frac{e e + 1}{2}$

Step 7.4.4

Multiply $e e$ .

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Step 7.4.4.1

Raise $e$ to the power of $1$ .

$\frac{e^{1} e + 1}{2}$

Step 7.4.4.2

Raise $e$ to the power of $1$ .

$\frac{e^{1} e^{1} + 1}{2}$

Step 7.4.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{e^{1 + 1} + 1}{2}$

Step 7.4.4.4

Add $1$ and $1$ .

$\frac{e^{2} + 1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{e^{2} + 1}{2}$

Decimal Form:

$4.19452804 \dots$

Step 9