Calculus Examples

$\frac{1}{9} \cdot \ln ({(x + 2)}^{9}) + \frac{1}{2} \cdot (\ln (x) - \ln ({(x^{2} + 3 x + 2)}^{2}))$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{9} \ln ({(x + 2)}^{9})$ by moving $\frac{1}{9}$ inside the logarithm.

$\ln ({({(x + 2)}^{9})}^{\frac{1}{9}}) + \frac{1}{2} \cdot (\ln (x) - \ln ({(x^{2} + 3 x + 2)}^{2}))$

Step 1.2

Multiply the exponents in ${({(x + 2)}^{9})}^{\frac{1}{9}}$ .

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Step 1.2.1

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

$\ln ({(x + 2)}^{9 (\frac{1}{9})}) + \frac{1}{2} \cdot (\ln (x) - \ln ({(x^{2} + 3 x + 2)}^{2}))$

Step 1.2.2

Cancel the common factor of $9$ .

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Step 1.2.2.1

Cancel the common factor.

$\ln ({(x + 2)}^{9 (\frac{1}{9})}) + \frac{1}{2} \cdot (\ln (x) - \ln ({(x^{2} + 3 x + 2)}^{2}))$

Step 1.2.2.2

Rewrite the expression.

$\ln ({(x + 2)}^{1}) + \frac{1}{2} \cdot (\ln (x) - \ln ({(x^{2} + 3 x + 2)}^{2}))$

Step 1.3

Simplify.

$\ln (x + 2) + \frac{1}{2} \cdot (\ln (x) - \ln ({(x^{2} + 3 x + 2)}^{2}))$

Step 1.4

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (x + 2) + \frac{1}{2} \cdot \ln (\frac{x}{{(x^{2} + 3 x + 2)}^{2}})$

Step 1.5

Simplify the denominator.

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Step 1.5.1

Factor $x^{2} + 3 x + 2$ using the AC method.

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Step 1.5.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $3$ .

$1, 2$

Step 1.5.1.2

Write the factored form using these integers.

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

Step 1.5.2

Apply the product rule to $(x + 1) (x + 2)$ .

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

Step 1.6

Simplify $\frac{1}{2} \ln (\frac{x}{{(x + 1)}^{2} {(x + 2)}^{2}})$ by moving $\frac{1}{2}$ inside the logarithm.

$\ln (x + 2) + \ln ({(\frac{x}{{(x + 1)}^{2} {(x + 2)}^{2}})}^{\frac{1}{2}})$

Step 1.7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.7.1

Apply the product rule to $\frac{x}{{(x + 1)}^{2} {(x + 2)}^{2}}$ .

$\ln (x + 2) + \ln (\frac{x^{\frac{1}{2}}}{{({(x + 1)}^{2} {(x + 2)}^{2})}^{\frac{1}{2}}})$

Step 1.7.2

Apply the product rule to ${(x + 1)}^{2} {(x + 2)}^{2}$ .

$\ln (x + 2) + \ln (\frac{x^{\frac{1}{2}}}{{({(x + 1)}^{2})}^{\frac{1}{2}} {({(x + 2)}^{2})}^{\frac{1}{2}}})$

Step 1.8

Simplify the denominator.

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Step 1.8.1

Multiply the exponents in ${({(x + 1)}^{2})}^{\frac{1}{2}}$ .

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Step 1.8.1.1

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

$\ln (x + 2) + \ln (\frac{x^{\frac{1}{2}}}{{(x + 1)}^{2 (\frac{1}{2})} {({(x + 2)}^{2})}^{\frac{1}{2}}})$

Step 1.8.1.2

Cancel the common factor of $2$ .

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Step 1.8.1.2.1

Cancel the common factor.

$\ln (x + 2) + \ln (\frac{x^{\frac{1}{2}}}{{(x + 1)}^{2 (\frac{1}{2})} {({(x + 2)}^{2})}^{\frac{1}{2}}})$

Step 1.8.1.2.2

Rewrite the expression.

$\ln (x + 2) + \ln (\frac{x^{\frac{1}{2}}}{{(x + 1)}^{1} {({(x + 2)}^{2})}^{\frac{1}{2}}})$

Step 1.8.2

Simplify.

$\ln (x + 2) + \ln (\frac{x^{\frac{1}{2}}}{(x + 1) {({(x + 2)}^{2})}^{\frac{1}{2}}})$

Step 1.8.3

Multiply the exponents in ${({(x + 2)}^{2})}^{\frac{1}{2}}$ .

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Step 1.8.3.1

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

$\ln (x + 2) + \ln (\frac{x^{\frac{1}{2}}}{(x + 1) {(x + 2)}^{2 (\frac{1}{2})}})$

Step 1.8.3.2

Cancel the common factor of $2$ .

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Step 1.8.3.2.1

Cancel the common factor.

$\ln (x + 2) + \ln (\frac{x^{\frac{1}{2}}}{(x + 1) {(x + 2)}^{2 (\frac{1}{2})}})$

Step 1.8.3.2.2

Rewrite the expression.

$\ln (x + 2) + \ln (\frac{x^{\frac{1}{2}}}{(x + 1) {(x + 2)}^{1}})$

Step 1.8.4

Simplify.

$\ln (x + 2) + \ln (\frac{x^{\frac{1}{2}}}{(x + 1) (x + 2)})$

Step 2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln ((x + 2) \frac{x^{\frac{1}{2}}}{(x + 1) (x + 2)})$

Step 3

Cancel the common factor of $x + 2$ .

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Step 3.1

Factor $x + 2$ out of $(x + 1) (x + 2)$ .

$\ln ((x + 2) \frac{x^{\frac{1}{2}}}{(x + 2) (x + 1)})$

Step 3.2

Cancel the common factor.

$\ln ((x + 2) \frac{x^{\frac{1}{2}}}{(x + 2) (x + 1)})$

Step 3.3

Rewrite the expression.

$\ln (\frac{x^{\frac{1}{2}}}{x + 1})$