Calculus Examples

$\int_{1}^{3} p \ln^{2} (x) d x$

Step 1

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

$p \int_{1}^{3} \ln^{2} (x) d x$

Step 2

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

$p (\ln^{2} (x) x]_{1}^{3} - \int_{1}^{3} x \frac{\ln (x^{2})}{x} d x)$

Step 3

Simplify.

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Step 3.1

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

$p (\ln^{2} (x) x]_{1}^{3} - \int_{1}^{3} \frac{x \ln (x^{2})}{x} d x)$

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

Cancel the common factor.

$p (\ln^{2} (x) x]_{1}^{3} - \int_{1}^{3} \frac{x \ln (x^{2})}{x} d x)$

Step 3.2.2

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

$p (\ln^{2} (x) x]_{1}^{3} - \int_{1}^{3} \ln (x^{2}) d x)$

Step 4

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

$p (\ln^{2} (x) x]_{1}^{3} - \int_{1}^{3} 2 \ln (x) d x)$

Step 5

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

$p (\ln^{2} (x) x]_{1}^{3} - (2 \int_{1}^{3} \ln (x) d x))$

Step 6

Multiply $2$ by $- 1$ .

$p (\ln^{2} (x) x]_{1}^{3} - 2 \int_{1}^{3} \ln (x) d x)$

Step 7

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

$p (\ln^{2} (x) x]_{1}^{3} - 2 (\ln (x) x]_{1}^{3} - \int_{1}^{3} x \frac{1}{x} d x))$

Step 8

Simplify.

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Step 8.1

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

$p (\ln^{2} (x) x]_{1}^{3} - 2 (\ln (x) x]_{1}^{3} - \int_{1}^{3} \frac{x}{x} d x))$

Step 8.2

Cancel the common factor of $x$ .

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Step 8.2.1

Cancel the common factor.

$p (\ln^{2} (x) x]_{1}^{3} - 2 (\ln (x) x]_{1}^{3} - \int_{1}^{3} \frac{x}{x} d x))$

Step 8.2.2

Rewrite the expression.

$p (\ln^{2} (x) x]_{1}^{3} - 2 (\ln (x) x]_{1}^{3} - \int_{1}^{3} d x))$

Step 9

Apply the constant rule.

$p (\ln^{2} (x) x]_{1}^{3} - 2 (\ln (x) x]_{1}^{3} - (x]_{1}^{3})))$

Step 10

Substitute and simplify.

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Step 10.1

Evaluate $\ln^{2} (x) x$ at $3$ and at $1$ .

$p ((\ln^{2} (3) \cdot 3) - \ln^{2} (1) \cdot 1 - 2 (\ln (x) x]_{1}^{3} - (x]_{1}^{3})))$

Step 10.2

Evaluate $\ln (x) x$ at $3$ and at $1$ .

$p ((\ln^{2} (3) \cdot 3) - \ln^{2} (1) \cdot 1 - 2 ((\ln (3) \cdot 3) - \ln (1) \cdot 1 - (x]_{1}^{3})))$

Step 10.3

Evaluate $x$ at $3$ and at $1$ .

$p ((\ln^{2} (3) \cdot 3) - \ln^{2} (1) \cdot 1 - 2 ((\ln (3) \cdot 3) - \ln (1) \cdot 1 - ((3) - 1)))$

Step 10.4

Simplify.

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Step 10.4.1

Move $3$ to the left of $\ln^{2} (3)$ .

$p (3 \cdot \ln^{2} (3) - \ln^{2} (1) \cdot 1 - 2 ((\ln (3) \cdot 3) - \ln (1) \cdot 1 - ((3) - 1)))$

Step 10.4.2

Multiply $- 1$ by $1$ .

$p (3 \ln^{2} (3) - \ln^{2} (1) - 2 ((\ln (3) \cdot 3) - \ln (1) \cdot 1 - ((3) - 1)))$

Step 10.4.3

Move $3$ to the left of $\ln (3)$ .

$p (3 \ln^{2} (3) - \ln^{2} (1) - 2 (3 \cdot \ln (3) - \ln (1) \cdot 1 - ((3) - 1)))$

Step 10.4.4

Multiply $- 1$ by $1$ .

$p (3 \ln^{2} (3) - \ln^{2} (1) - 2 (3 \ln (3) - \ln (1) - ((3) - 1)))$

Step 10.4.5

Subtract $1$ from $3$ .

$p (3 \ln^{2} (3) - \ln^{2} (1) - 2 (3 \ln (3) - \ln (1) - 1 \cdot 2))$

Step 10.4.6

Multiply $- 1$ by $2$ .

$p (3 \ln^{2} (3) - \ln^{2} (1) - 2 (3 \ln (3) - \ln (1) - 2))$

Step 11

Simplify.

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Step 11.1

Simplify each term.

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Step 11.1.1

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

$p (3 \ln^{2} (3) - 0^{2} - 2 (3 \ln (3) - \ln (1) - 2))$

Step 11.1.2

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

$p (3 \ln^{2} (3) - 0 - 2 (3 \ln (3) - \ln (1) - 2))$

Step 11.1.3

Multiply $- 1$ by $0$ .

$p (3 \ln^{2} (3) + 0 - 2 (3 \ln (3) - \ln (1) - 2))$

Step 11.1.4

Simplify each term.

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Step 11.1.4.1

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

$p (3 \ln^{2} (3) + 0 - 2 (3 \ln (3) - 0 - 2))$

Step 11.1.4.2

Multiply $- 1$ by $0$ .

$p (3 \ln^{2} (3) + 0 - 2 (3 \ln (3) + 0 - 2))$

Step 11.1.5

Add $3 \ln (3)$ and $0$ .

$p (3 \ln^{2} (3) + 0 - 2 (3 \ln (3) - 2))$

Step 11.1.6

Apply the distributive property.

$p (3 \ln^{2} (3) + 0 - 2 (3 \ln (3)) - 2 \cdot - 2)$

Step 11.1.7

Multiply $3$ by $- 2$ .

$p (3 \ln^{2} (3) + 0 - 6 \ln (3) - 2 \cdot - 2)$

Step 11.1.8

Multiply $- 2$ by $- 2$ .

$p (3 \ln^{2} (3) + 0 - 6 \ln (3) + 4)$

Step 11.2

Add $3 \ln^{2} (3)$ and $0$ .

$p (3 \ln^{2} (3) - 6 \ln (3) + 4)$

Step 11.3

Apply the distributive property.

$p (3 \ln^{2} (3)) + p (- 6 \ln (3)) + p \cdot 4$

Step 11.4

Move $4$ to the left of $p$ .

$p \cdot 3 \ln^{2} (3) + p (- 6 \ln (3)) + 4 \cdot p$

Step 11.5

Move $3$ to the left of $p$ .

$3 p \ln^{2} (3) + p (- 6 \ln (3)) + 4 p$