Calculus Examples

$\int_{1}^{7} \frac{\ln ({(x)}^{2})}{x^{3}} d x$

Step 1

Rewrite $\frac{\ln (x^{2})}{x^{3}}$ as $\frac{2 \ln (x)}{x^{3}}$ .

$\int_{1}^{7} \frac{2 \ln (x)}{x^{3}} d x$

Step 2

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

$2 \int_{1}^{7} \frac{\ln (x)}{x^{3}} d x$

Step 3

Apply basic rules of exponents.

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Step 3.1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$2 \int_{1}^{7} \ln (x) {(x^{3})}^{- 1} d x$

Step 3.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 \int_{1}^{7} \ln (x) x^{3 \cdot - 1} d x$

Step 3.2.2

Multiply $3$ by $- 1$ .

$2 \int_{1}^{7} \ln (x) x^{- 3} d x$

Step 4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = x^{- 3}$ .

$2 (\ln (x) (- \frac{1}{2 x^{2}})]_{1}^{7} - \int_{1}^{7} - \frac{1}{2 x^{2}} \cdot \frac{1}{x} d x)$

Step 5

Simplify.

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Step 5.1

Combine $\ln (x)$ and $\frac{1}{2 x^{2}}$ .

$2 (- \frac{\ln (x)}{2 x^{2}}]_{1}^{7} - \int_{1}^{7} - \frac{1}{2 x^{2}} \cdot \frac{1}{x} d x)$

Step 5.2

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

$2 (- \frac{\ln (x)}{2 x^{2}}]_{1}^{7} - \int_{1}^{7} - \frac{1}{x (2 x^{2})} d x)$

Step 5.3

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

$2 (- \frac{\ln (x)}{2 x^{2}}]_{1}^{7} - \int_{1}^{7} - \frac{1}{2 (x^{1} x^{2})} d x)$

Step 5.4

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

$2 (- \frac{\ln (x)}{2 x^{2}}]_{1}^{7} - \int_{1}^{7} - \frac{1}{2 x^{1 + 2}} d x)$

Step 5.5

Add $1$ and $2$ .

$2 (- \frac{\ln (x)}{2 x^{2}}]_{1}^{7} - \int_{1}^{7} - \frac{1}{2 x^{3}} d x)$

Step 6

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$2 (- \frac{\ln (x)}{2 x^{2}}]_{1}^{7} - - \int_{1}^{7} \frac{1}{2 x^{3}} d x)$

Step 7

Simplify.

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Step 7.1

Multiply $- 1$ by $- 1$ .

$2 (- \frac{\ln (x)}{2 x^{2}}]_{1}^{7} + 1 \int_{1}^{7} \frac{1}{2 x^{3}} d x)$

Step 7.2

Multiply $\int_{1}^{7} \frac{1}{2 x^{3}} d x$ by $1$ .

$2 (- \frac{\ln (x)}{2 x^{2}}]_{1}^{7} + \int_{1}^{7} \frac{1}{2 x^{3}} d x)$

Step 8

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$2 (- \frac{\ln (x)}{2 x^{2}}]_{1}^{7} + \frac{1}{2} \int_{1}^{7} \frac{1}{x^{3}} d x)$

Step 9

Apply basic rules of exponents.

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Step 9.1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$2 (- \frac{\ln (x)}{2 x^{2}}]_{1}^{7} + \frac{1}{2} \int_{1}^{7} {(x^{3})}^{- 1} d x)$

Step 9.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 9.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2 (- \frac{\ln (x)}{2 x^{2}}]_{1}^{7} + \frac{1}{2} \int_{1}^{7} x^{3 \cdot - 1} d x)$

Step 9.2.2

Multiply $3$ by $- 1$ .

$2 (- \frac{\ln (x)}{2 x^{2}}]_{1}^{7} + \frac{1}{2} \int_{1}^{7} x^{- 3} d x)$

Step 10

By the Power Rule, the integral of $x^{- 3}$ with respect to $x$ is $- \frac{1}{2} x^{- 2}$ .

$2 (- \frac{\ln (x)}{2 x^{2}}]_{1}^{7} + \frac{1}{2} - \frac{1}{2} x^{- 2}]_{1}^{7})$

Step 11

Simplify the answer.

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Step 11.1

Simplify.

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Step 11.1.1

Combine $x^{- 2}$ and $\frac{1}{2}$ .

$2 (- \frac{\ln (x)}{2 x^{2}}]_{1}^{7} + \frac{1}{2} - \frac{x^{- 2}}{2}]_{1}^{7})$

Step 11.1.2

Combine $\frac{1}{2}$ and $- \frac{x^{- 2}}{2}]_{1}^{7}$ .

$2 (- \frac{\ln (x)}{2 x^{2}}]_{1}^{7} + \frac{- \frac{x^{- 2}}{2}]_{1}^{7}}{2})$

Step 11.1.3

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 (- \frac{\ln (x)}{2 x^{2}}]_{1}^{7} + \frac{- \frac{1}{2 x^{2}}]_{1}^{7}}{2})$

Step 11.2

Substitute and simplify.

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Step 11.2.1

Evaluate $- \frac{\ln (x)}{2 x^{2}}$ at $7$ and at $1$ .

$2 ((- \frac{\ln (7)}{2 \cdot 7^{2}}) + \frac{\ln (1)}{2 \cdot 1^{2}} + \frac{- \frac{1}{2 x^{2}}]_{1}^{7}}{2})$

Step 11.2.2

Evaluate $- \frac{1}{2 x^{2}}$ at $7$ and at $1$ .

$2 ((- \frac{\ln (7)}{2 \cdot 7^{2}}) + \frac{\ln (1)}{2 \cdot 1^{2}} + \frac{(- \frac{1}{2 \cdot 7^{2}}) + \frac{1}{2 \cdot 1^{2}}}{2})$

Step 11.2.3

Simplify.

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Step 11.2.3.1

Raise $7$ to the power of $2$ .

$2 (- \frac{\ln (7)}{2 \cdot 49} + \frac{\ln (1)}{2 \cdot 1^{2}} + \frac{(- \frac{1}{2 \cdot 7^{2}}) + \frac{1}{2 \cdot 1^{2}}}{2})$

Step 11.2.3.2

Multiply $2$ by $49$ .

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2 \cdot 1^{2}} + \frac{(- \frac{1}{2 \cdot 7^{2}}) + \frac{1}{2 \cdot 1^{2}}}{2})$

Step 11.2.3.3

One to any power is one.

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2 \cdot 1} + \frac{(- \frac{1}{2 \cdot 7^{2}}) + \frac{1}{2 \cdot 1^{2}}}{2})$

Step 11.2.3.4

Multiply $2$ by $1$ .

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2} + \frac{(- \frac{1}{2 \cdot 7^{2}}) + \frac{1}{2 \cdot 1^{2}}}{2})$

Step 11.2.3.5

Raise $7$ to the power of $2$ .

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2} + \frac{- \frac{1}{2 \cdot 49} + \frac{1}{2 \cdot 1^{2}}}{2})$

Step 11.2.3.6

Multiply $2$ by $49$ .

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2} + \frac{- \frac{1}{98} + \frac{1}{2 \cdot 1^{2}}}{2})$

Step 11.2.3.7

One to any power is one.

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2} + \frac{- \frac{1}{98} + \frac{1}{2 \cdot 1}}{2})$

Step 11.2.3.8

Multiply $2$ by $1$ .

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2} + \frac{- \frac{1}{98} + \frac{1}{2}}{2})$

Step 11.2.3.9

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{49}{49}$ .

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2} + \frac{- \frac{1}{98} + \frac{1}{2} \cdot \frac{49}{49}}{2})$

Step 11.2.3.10

Write each expression with a common denominator of $98$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.2.3.10.1

Multiply $\frac{1}{2}$ by $\frac{49}{49}$ .

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2} + \frac{- \frac{1}{98} + \frac{49}{2 \cdot 49}}{2})$

Step 11.2.3.10.2

Multiply $2$ by $49$ .

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2} + \frac{- \frac{1}{98} + \frac{49}{98}}{2})$

Step 11.2.3.11

Combine the numerators over the common denominator.

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2} + \frac{\frac{- 1 + 49}{98}}{2})$

Step 11.2.3.12

Add $- 1$ and $49$ .

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2} + \frac{\frac{48}{98}}{2})$

Step 11.2.3.13

Cancel the common factor of $48$ and $98$ .

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Step 11.2.3.13.1

Factor $2$ out of $48$ .

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2} + \frac{\frac{2 (24)}{98}}{2})$

Step 11.2.3.13.2

Cancel the common factors.

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Step 11.2.3.13.2.1

Factor $2$ out of $98$ .

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2} + \frac{\frac{2 \cdot 24}{2 \cdot 49}}{2})$

Step 11.2.3.13.2.2

Cancel the common factor.

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2} + \frac{\frac{2 \cdot 24}{2 \cdot 49}}{2})$

Step 11.2.3.13.2.3

Rewrite the expression.

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2} + \frac{\frac{24}{49}}{2})$

Step 11.2.3.14

Rewrite $\frac{\frac{24}{49}}{2}$ as a product.

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2} + \frac{24}{49} \cdot \frac{1}{2})$

Step 11.2.3.15

Multiply $\frac{24}{49}$ by $\frac{1}{2}$ .

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2} + \frac{24}{49 \cdot 2})$

Step 11.2.3.16

Multiply $49$ by $2$ .

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2} + \frac{24}{98})$

Step 11.2.3.17

Cancel the common factor of $24$ and $98$ .

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Step 11.2.3.17.1

Factor $2$ out of $24$ .

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2} + \frac{2 (12)}{98})$

Step 11.2.3.17.2

Cancel the common factors.

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Step 11.2.3.17.2.1

Factor $2$ out of $98$ .

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2} + \frac{2 \cdot 12}{2 \cdot 49})$

Step 11.2.3.17.2.2

Cancel the common factor.

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2} + \frac{2 \cdot 12}{2 \cdot 49})$

Step 11.2.3.17.2.3

Rewrite the expression.

$2 (- \frac{\ln (7)}{98} + \frac{\ln (1)}{2} + \frac{12}{49})$

Step 12

Simplify.

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Step 12.1

Simplify each term.

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Step 12.1.1

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

$2 (- \frac{\ln (7)}{98} + \frac{0}{2} + \frac{12}{49})$

Step 12.1.2

Divide $0$ by $2$ .

$2 (- \frac{\ln (7)}{98} + 0 + \frac{12}{49})$

Step 12.2

Add $- \frac{\ln (7)}{98}$ and $0$ .

$2 (- \frac{\ln (7)}{98} + \frac{12}{49})$

Step 12.3

Apply the distributive property.

$2 (- \frac{\ln (7)}{98}) + 2 (\frac{12}{49})$

Step 12.4

Cancel the common factor of $2$ .

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Step 12.4.1

Move the leading negative in $- \frac{\ln (7)}{98}$ into the numerator.

$2 \frac{- \ln (7)}{98} + 2 (\frac{12}{49})$

Step 12.4.2

Factor $2$ out of $98$ .

$2 \frac{- \ln (7)}{2 (49)} + 2 (\frac{12}{49})$

Step 12.4.3

Cancel the common factor.

$2 \frac{- \ln (7)}{2 \cdot 49} + 2 (\frac{12}{49})$

Step 12.4.4

Rewrite the expression.

$\frac{- \ln (7)}{49} + 2 (\frac{12}{49})$

Step 12.5

Multiply $2 (\frac{12}{49})$ .

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Step 12.5.1

Combine $2$ and $\frac{12}{49}$ .

$\frac{- \ln (7)}{49} + \frac{2 \cdot 12}{49}$

Step 12.5.2

Multiply $2$ by $12$ .

$\frac{- \ln (7)}{49} + \frac{24}{49}$

Step 12.6

Move the negative in front of the fraction.

$- \frac{\ln (7)}{49} + \frac{24}{49}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$- \frac{\ln (7)}{49} + \frac{24}{49}$

Decimal Form:

$0.45008346 \dots$