Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Cancel the common factor of $x$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

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

Step 5

Apply the constant rule.

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

Step 6

Substitute and simplify.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Multiply $- 1$ by $1$ .

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

Step 7

Simplify.

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Step 7.1

Simplify each term.

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Step 7.1.1

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

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

Step 7.1.2

Multiply $e$ by $1$ .

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

Step 7.1.3

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

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

Step 7.1.4

Multiply $- 1$ by $0$ .

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

Step 7.1.5

Apply the distributive property.

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

Step 7.1.6

Multiply $- 1$ by $- 1$ .

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

Step 7.2

Add $e$ and $0$ .

$2 (e - e + 1)$

Step 7.3

Subtract $e$ from $e$ .

$2 (0 + 1)$

Step 7.4

Add $0$ and $1$ .

$2 \cdot 1$

Step 7.5

Multiply $2$ by $1$ .

$2$