Calculus Examples

$\int_{e}^{e^{2}} \ln (x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = 1$ .

$\ln (x) x]_{e}^{e^{2}} - \int_{e}^{e^{2}} x \frac{1}{x} d x$

Step 2

Simplify.

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Step 2.1

Combine $x$ and $\frac{1}{x}$ .

$\ln (x) x]_{e}^{e^{2}} - \int_{e}^{e^{2}} \frac{x}{x} d x$

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

$\ln (x) x]_{e}^{e^{2}} - \int_{e}^{e^{2}} \frac{x}{x} d x$

Step 2.2.2

Rewrite the expression.

$\ln (x) x]_{e}^{e^{2}} - \int_{e}^{e^{2}} d x$

Step 3

Apply the constant rule.

$\ln (x) x]_{e}^{e^{2}} - (x]_{e}^{e^{2}})$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $\ln (x) x$ at $e^{2}$ and at $e$ .

$(\ln (e^{2}) e^{2}) - \ln (e) e - (x]_{e}^{e^{2}})$

Step 4.2

Evaluate $x$ at $e^{2}$ and at $e$ .

$(\ln (e^{2}) e^{2}) - \ln (e) e - ((e^{2}) - e)$

Step 4.3

Simplify.

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Step 4.3.1

Use logarithm rules to move $2$ out of the exponent.

$2 \ln (e) e^{2} - \ln (e) e - ((e^{2}) - e)$

Step 4.3.2

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

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

Step 4.3.3

Multiply $2$ by $1$ .

$2 e^{2} - \ln (e) e - (e^{2} - e)$

Step 5

Simplify.

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Step 5.1

Simplify each term.

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Step 5.1.1

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

$2 e^{2} - 1 \cdot 1 e - (e^{2} - e)$

Step 5.1.2

Multiply $- 1$ by $1$ .

$2 e^{2} - e - (e^{2} - e)$

Step 5.1.3

Apply the distributive property.

$2 e^{2} - e - e^{2} - - e$

Step 5.1.4

Multiply $- - e$ .

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Step 5.1.4.1

Multiply $- 1$ by $- 1$ .

$2 e^{2} - e - e^{2} + 1 e$

Step 5.1.4.2

Multiply $e$ by $1$ .

$2 e^{2} - e - e^{2} + e$

Step 5.2

Subtract $e^{2}$ from $2 e^{2}$ .

$e^{2} - e + e$

Step 5.3

Add $- e$ and $e$ .

$e^{2} + 0$

Step 5.4

Add $e^{2}$ and $0$ .

$e^{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$e^{2}$

Decimal Form:

$7.38905609 \dots$