Calculus Examples

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

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln^{2} (x)$ and $d v = 1$ .

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

Step 2

Simplify.

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Step 2.1

Combine $x$ and $\frac{\ln (x^{2})}{x}$ .

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

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Divide $\ln (x^{2})$ by $1$ .

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

Step 3

Rewrite $\ln (x^{2})$ as $2 \ln (x)$ .

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

Step 4

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 5

Multiply $2$ by $- 1$ .

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

Cancel the common factor of $x$ .

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

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

Step 8

Apply the constant rule.

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

Step 9

Substitute and simplify.

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Step 9.1

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

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

Step 9.2

Evaluate $\ln (x) x$ at $e$ and at $1$ .

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

Step 9.3

Evaluate $x$ at $e$ and at $1$ .

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

Step 9.4

Simplify.

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Step 9.4.1

Multiply $- 1$ by $1$ .

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

Step 9.4.2

Multiply $- 1$ by $1$ .

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

Step 10

Simplify.

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Step 10.1

Simplify each term.

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Step 10.1.1

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

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

Step 10.1.2

One to any power is one.

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

Step 10.1.3

Multiply $e$ by $1$ .

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

Step 10.1.4

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

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

Step 10.1.5

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

$e - 0 - 2 (\ln (e) e - \ln (1) - (e - 1))$

Step 10.1.6

Multiply $- 1$ by $0$ .

$e + 0 - 2 (\ln (e) e - \ln (1) - (e - 1))$

Step 10.1.7

Simplify each term.

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Step 10.1.7.1

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

$e + 0 - 2 (1 e - \ln (1) - (e - 1))$

Step 10.1.7.2

Multiply $e$ by $1$ .

$e + 0 - 2 (e - \ln (1) - (e - 1))$

Step 10.1.7.3

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

$e + 0 - 2 (e - 0 - (e - 1))$

Step 10.1.7.4

Multiply $- 1$ by $0$ .

$e + 0 - 2 (e + 0 - (e - 1))$

Step 10.1.7.5

Apply the distributive property.

$e + 0 - 2 (e + 0 - e - - 1)$

Step 10.1.7.6

Multiply $- 1$ by $- 1$ .

$e + 0 - 2 (e + 0 - e + 1)$

Step 10.1.8

Add $e$ and $0$ .

$e + 0 - 2 (e - e + 1)$

Step 10.1.9

Subtract $e$ from $e$ .

$e + 0 - 2 (0 + 1)$

Step 10.1.10

Add $0$ and $1$ .

$e + 0 - 2 \cdot 1$

Step 10.1.11

Multiply $- 2$ by $1$ .

$e + 0 - 2$

Step 10.2

Add $e$ and $0$ .

$e - 2$

Step 11

The result can be shown in multiple forms.

Exact Form:

$e - 2$

Decimal Form:

$0.71828182 \dots$