Calculus Examples

$\int_{0}^{1} x^{2} \cdot 2^{x} d x$

Step 1

Integrate by parts using the formula

$\int u d v = u v - \int v d u$ , where

$u = x^{2}$ and

$d v = 2^{x}$ .

$x^{2} (\frac{1}{\ln (2)} \cdot 2^{x})]_{0}^{1} - \int_{0}^{1} \frac{1}{\ln (2)} \cdot 2^{x} (2 x) d x$

Step 2

Simplify.

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Step 2.1

Combine

$\frac{1}{\ln (2)}$ and

$2^{x}$ .

$x^{2} \frac{2^{x}}{\ln (2)}]_{0}^{1} - \int_{0}^{1} \frac{1}{\ln (2)} \cdot 2^{x} (2 x) d x$

Step 2.2

Combine

$x^{2}$ and

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

$\frac{x^{2} \cdot 2^{x}}{\ln (2)}]_{0}^{1} - \int_{0}^{1} \frac{1}{\ln (2)} \cdot 2^{x} (2 x) d x$

Step 3

Since

$\frac{1}{\ln (2)} \cdot 2$ is constant with respect to

$x$ , move

$\frac{1}{\ln (2)} \cdot 2$ out of the integral.

$\frac{x^{2} \cdot 2^{x}}{\ln (2)}]_{0}^{1} - (\frac{1}{\ln (2)} \cdot 2 \int_{0}^{1} 2^{x - 1} (2 x) d x)$

Step 4

Simplify.

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Step 4.1

Combine

$\frac{1}{\ln (2)}$ and

$2$ .

$\frac{x^{2} \cdot 2^{x}}{\ln (2)}]_{0}^{1} - (\frac{2}{\ln (2)} \int_{0}^{1} 2^{x - 1} (2 x) d x)$

Step 4.2

Multiply

$2^{x - 1}$ by

$2$ by adding the exponents.

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Step 4.2.1

Move

$2$ .

$\frac{x^{2} \cdot 2^{x}}{\ln (2)}]_{0}^{1} - (\frac{2}{\ln (2)} \int_{0}^{1} 2 \cdot 2^{x - 1} x d x)$

Step 4.2.2

Multiply

$2$ by

$2^{x - 1}$ .

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Step 4.2.2.1

Raise

$2$ to the power of

$1$ .

$\frac{x^{2} \cdot 2^{x}}{\ln (2)}]_{0}^{1} - (\frac{2}{\ln (2)} \int_{0}^{1} 2^{1} \cdot 2^{x - 1} x d x)$

Step 4.2.2.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{2} \cdot 2^{x}}{\ln (2)}]_{0}^{1} - (\frac{2}{\ln (2)} \int_{0}^{1} 2^{1 + x - 1} x d x)$

Step 4.2.3

Combine the opposite terms in

$1 + x - 1$ .

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Step 4.2.3.1

Subtract

$1$ from

$1$ .

$\frac{x^{2} \cdot 2^{x}}{\ln (2)}]_{0}^{1} - (\frac{2}{\ln (2)} \int_{0}^{1} 2^{x + 0} x d x)$

Step 4.2.3.2

Add

$x$ and

$0$ .

$\frac{x^{2} \cdot 2^{x}}{\ln (2)}]_{0}^{1} - (\frac{2}{\ln (2)} \int_{0}^{1} 2^{x} x d x)$

$\frac{x^{2} \cdot 2^{x}}{\ln (2)}]_{0}^{1} - \frac{2}{\ln (2)} \int_{0}^{1} 2^{x} x d x$

Step 5

Integrate by parts using the formula

$\int u d v = u v - \int v d u$ , where

$u = x$ and

$d v = 2^{x}$ .

$\frac{x^{2} \cdot 2^{x}}{\ln (2)}]_{0}^{1} - \frac{2}{\ln (2)} (x (\frac{1}{\ln (2)} \cdot 2^{x})]_{0}^{1} - \int_{0}^{1} \frac{1}{\ln (2)} \cdot 2^{x} d x)$

Step 6

Simplify.

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Step 6.1

Combine

$\frac{1}{\ln (2)}$ and

$2^{x}$ .

$\frac{x^{2} \cdot 2^{x}}{\ln (2)}]_{0}^{1} - \frac{2}{\ln (2)} (x \frac{2^{x}}{\ln (2)}]_{0}^{1} - \int_{0}^{1} \frac{1}{\ln (2)} \cdot 2^{x} d x)$

Step 6.2

Combine

$x$ and

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

$\frac{x^{2} \cdot 2^{x}}{\ln (2)}]_{0}^{1} - \frac{2}{\ln (2)} (\frac{x \cdot 2^{x}}{\ln (2)}]_{0}^{1} - \int_{0}^{1} \frac{1}{\ln (2)} \cdot 2^{x} d x)$

Step 6.3

Combine

$\frac{1}{\ln (2)}$ and

$2^{x}$ .

$\frac{x^{2} \cdot 2^{x}}{\ln (2)}]_{0}^{1} - \frac{2}{\ln (2)} (\frac{x \cdot 2^{x}}{\ln (2)}]_{0}^{1} - \int_{0}^{1} \frac{2^{x}}{\ln (2)} d x)$

Step 7

Since

$\frac{1}{\ln (2)}$ is constant with respect to

$x$ , move

$\frac{1}{\ln (2)}$ out of the integral.

$\frac{x^{2} \cdot 2^{x}}{\ln (2)}]_{0}^{1} - \frac{2}{\ln (2)} (\frac{x \cdot 2^{x}}{\ln (2)}]_{0}^{1} - (\frac{1}{\ln (2)} \int_{0}^{1} 2^{x} d x))$

Step 8

The integral of

$2^{x}$ with respect to

$x$ is

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

$\frac{x^{2} \cdot 2^{x}}{\ln (2)}]_{0}^{1} - \frac{2}{\ln (2)} (\frac{x \cdot 2^{x}}{\ln (2)}]_{0}^{1} - \frac{1}{\ln (2)} \frac{2^{x}}{\ln (2)}]_{0}^{1})$

Step 9

Simplify the answer.

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Step 9.1

Combine

$\frac{2^{x}}{\ln (2)}]_{0}^{1}$ and

$\frac{1}{\ln (2)}$ .

$\frac{x^{2} \cdot 2^{x}}{\ln (2)}]_{0}^{1} - \frac{2}{\ln (2)} (\frac{x \cdot 2^{x}}{\ln (2)}]_{0}^{1} - \frac{\frac{2^{x}}{\ln (2)}]_{0}^{1}}{\ln (2)})$

Step 9.2

Substitute and simplify.

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Step 9.2.1

Evaluate

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

$1$ and at

$0$ .

$(\frac{1^{2} \cdot 2^{1}}{\ln (2)}) - \frac{0^{2} \cdot 2^{0}}{\ln (2)} - \frac{2}{\ln (2)} (\frac{x \cdot 2^{x}}{\ln (2)}]_{0}^{1} - \frac{\frac{2^{x}}{\ln (2)}]_{0}^{1}}{\ln (2)})$

Step 9.2.2

Evaluate

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

$1$ and at

$0$ .

$(\frac{1^{2} \cdot 2^{1}}{\ln (2)}) - \frac{0^{2} \cdot 2^{0}}{\ln (2)} - \frac{2}{\ln (2)} ((\frac{1 \cdot 2^{1}}{\ln (2)}) - \frac{0 \cdot 2^{0}}{\ln (2)} - \frac{\frac{2^{x}}{\ln (2)}]_{0}^{1}}{\ln (2)})$

Step 9.2.3

Evaluate

$\frac{2^{x}}{\ln (2)}$ at

$1$ and at

$0$ .

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

Step 9.2.4

Simplify.

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Step 9.2.4.1

One to any power is one.

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

Step 9.2.4.2

Evaluate the exponent.

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

Step 9.2.4.3

Multiply

$2$ by

$1$ .

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

Step 9.2.4.4

Raising

$0$ to any positive power yields

$0$ .

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

Step 9.2.4.5

Anything raised to

$0$ is

$1$ .

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

Step 9.2.4.6

Multiply

$0$ by

$1$ .

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

Step 9.2.4.7

Combine the numerators over the common denominator.

$\frac{2 + 0}{\ln (2)} - \frac{2}{\ln (2)} ((\frac{1 \cdot 2^{1}}{\ln (2)}) - \frac{0 \cdot 2^{0}}{\ln (2)} - \frac{(\frac{2^{1}}{\ln (2)}) - \frac{2^{0}}{\ln (2)}}{\ln (2)})$

Step 9.2.4.8

Add

$2$ and

$0$ .

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

Step 9.2.4.9

Evaluate the exponent.

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

Step 9.2.4.10

Multiply

$2$ by

$1$ .

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

Step 9.2.4.11

Anything raised to

$0$ is

$1$ .

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

Step 9.2.4.12

Multiply

$0$ by

$1$ .

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

Step 9.2.4.13

Combine the numerators over the common denominator.

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

Step 9.2.4.14

Add

$2$ and

$0$ .

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

Step 9.2.4.15

Evaluate the exponent.

$\frac{2}{\ln (2)} - \frac{2}{\ln (2)} (\frac{2}{\ln (2)} - \frac{\frac{2}{\ln (2)} - \frac{2^{0}}{\ln (2)}}{\ln (2)})$

Step 9.2.4.16

Anything raised to

$0$ is

$1$ .

$\frac{2}{\ln (2)} - \frac{2}{\ln (2)} (\frac{2}{\ln (2)} - \frac{\frac{2}{\ln (2)} - \frac{1}{\ln (2)}}{\ln (2)})$

Step 9.2.4.17

Combine the numerators over the common denominator.

$\frac{2}{\ln (2)} - \frac{2}{\ln (2)} (\frac{2}{\ln (2)} - \frac{\frac{2 - 1}{\ln (2)}}{\ln (2)})$

Step 9.2.4.18

Subtract

$1$ from

$2$ .

$\frac{2}{\ln (2)} - \frac{2}{\ln (2)} (\frac{2}{\ln (2)} - \frac{\frac{1}{\ln (2)}}{\ln (2)})$

Step 9.2.4.19

Rewrite

$\frac{\frac{1}{\ln (2)}}{\ln (2)}$ as a product.

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

Step 9.2.4.20

Multiply

$\frac{1}{\ln (2)}$ by

$\frac{1}{\ln (2)}$ .

$\frac{2}{\ln (2)} - \frac{2}{\ln (2)} (\frac{2}{\ln (2)} - \frac{1}{\ln (2) \ln (2)})$

Step 9.2.4.21

Raise

$\ln (2)$ to the power of

$1$ .

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

Step 9.2.4.22

Raise

$\ln (2)$ to the power of

$1$ .

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

Step 9.2.4.23

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 9.2.4.24

Add

$1$ and

$1$ .

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

Step 9.2.4.25

To write

$- \frac{2}{\ln (2)} (\frac{2}{\ln (2)} - \frac{1}{\ln^{2} (2)})$ as a fraction with a common denominator, multiply by

$\frac{\ln (2)}{\ln (2)}$ .

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

Step 9.2.4.26

Combine

$- \frac{2}{\ln (2)} (\frac{2}{\ln (2)} - \frac{1}{\ln^{2} (2)})$ and

$\frac{\ln (2)}{\ln (2)}$ .

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

Step 9.2.4.27

Combine the numerators over the common denominator.

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

Step 9.2.4.28

Combine

$\ln (2)$ and

$\frac{2}{\ln (2)}$ .

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

Step 9.2.4.29

Move

$2$ to the left of

$\ln (2)$ .

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

Step 9.2.4.30

Cancel the common factor of

$\ln (2)$ .

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Step 9.2.4.30.1

Cancel the common factor.

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

Step 9.2.4.30.2

Divide

$2$ by

$1$ .

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

Step 9.2.4.31

Multiply

$- 1$ by

$2$ .

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

Step 10

Simplify.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Apply the distributive property.

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

Step 10.1.2

Multiply

$- 2 \frac{2}{\ln (2)}$ .

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Step 10.1.2.1

Combine

$- 2$ and

$\frac{2}{\ln (2)}$ .

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

Step 10.1.2.2

Multiply

$- 2$ by

$2$ .

$\frac{2 + \frac{- 4}{\ln (2)} - 2 (- \frac{1}{\ln^{2} (2)})}{\ln (2)}$

Step 10.1.3

Multiply

$- 2 (- \frac{1}{\ln^{2} (2)})$ .

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Step 10.1.3.1

Multiply

$- 1$ by

$- 2$ .

$\frac{2 + \frac{- 4}{\ln (2)} + 2 \frac{1}{\ln^{2} (2)}}{\ln (2)}$

Step 10.1.3.2

Combine

$2$ and

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

$\frac{2 + \frac{- 4}{\ln (2)} + \frac{2}{\ln^{2} (2)}}{\ln (2)}$

Step 10.1.4

Move the negative in front of the fraction.

$\frac{2 - \frac{4}{\ln (2)} + \frac{2}{\ln^{2} (2)}}{\ln (2)}$

Step 10.1.5

To write

$2$ as a fraction with a common denominator, multiply by

$\frac{\ln (2)}{\ln (2)}$ .

$\frac{\frac{2 \ln (2)}{\ln (2)} - \frac{4}{\ln (2)} + \frac{2}{\ln^{2} (2)}}{\ln (2)}$

Step 10.1.6

Combine the numerators over the common denominator.

$\frac{\frac{2 \ln (2) - 4}{\ln (2)} + \frac{2}{\ln^{2} (2)}}{\ln (2)}$

Step 10.1.7

To write

$\frac{2 \ln (2) - 4}{\ln (2)}$ as a fraction with a common denominator, multiply by

$\frac{\ln (2)}{\ln (2)}$ .

$\frac{\frac{2 \ln (2) - 4}{\ln (2)} \cdot \frac{\ln (2)}{\ln (2)} + \frac{2}{\ln^{2} (2)}}{\ln (2)}$

Step 10.1.8

Write each expression with a common denominator of

$\ln^{2} (2)$ , by multiplying each by an appropriate factor of

$1$ .

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Step 10.1.8.1

Multiply

$\frac{2 \ln (2) - 4}{\ln (2)}$ by

$\frac{\ln (2)}{\ln (2)}$ .

$\frac{\frac{(2 \ln (2) - 4) \ln (2)}{\ln (2) \ln (2)} + \frac{2}{\ln^{2} (2)}}{\ln (2)}$

Step 10.1.8.2

Raise

$\ln (2)$ to the power of

$1$ .

$\frac{\frac{(2 \ln (2) - 4) \ln (2)}{\ln^{1} (2) \ln (2)} + \frac{2}{\ln^{2} (2)}}{\ln (2)}$

Step 10.1.8.3

Raise

$\ln (2)$ to the power of

$1$ .

$\frac{\frac{(2 \ln (2) - 4) \ln (2)}{\ln^{1} (2) \ln^{1} (2)} + \frac{2}{\ln^{2} (2)}}{\ln (2)}$

Step 10.1.8.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{(2 \ln (2) - 4) \ln (2)}{{\ln (2)}^{1 + 1}} + \frac{2}{\ln^{2} (2)}}{\ln (2)}$

Step 10.1.8.5

Add

$1$ and

$1$ .

$\frac{\frac{(2 \ln (2) - 4) \ln (2)}{\ln^{2} (2)} + \frac{2}{\ln^{2} (2)}}{\ln (2)}$

Step 10.1.9

Combine the numerators over the common denominator.

$\frac{\frac{(2 \ln (2) - 4) \ln (2) + 2}{\ln^{2} (2)}}{\ln (2)}$

Step 10.1.10

Simplify the numerator.

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Step 10.1.10.1

Apply the distributive property.

$\frac{\frac{2 \ln (2) \ln (2) - 4 \ln (2) + 2}{\ln^{2} (2)}}{\ln (2)}$

Step 10.1.10.2

Multiply

$2 \ln (2) \ln (2)$ .

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Step 10.1.10.2.1

Raise

$\ln (2)$ to the power of

$1$ .

$\frac{\frac{2 (\ln^{1} (2) \ln (2)) - 4 \ln (2) + 2}{\ln^{2} (2)}}{\ln (2)}$

Step 10.1.10.2.2

Raise

$\ln (2)$ to the power of

$1$ .

$\frac{\frac{2 (\ln^{1} (2) \ln^{1} (2)) - 4 \ln (2) + 2}{\ln^{2} (2)}}{\ln (2)}$

Step 10.1.10.2.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{2 {\ln (2)}^{1 + 1} - 4 \ln (2) + 2}{\ln^{2} (2)}}{\ln (2)}$

Step 10.1.10.2.4

Add

$1$ and

$1$ .

$\frac{\frac{2 \ln^{2} (2) - 4 \ln (2) + 2}{\ln^{2} (2)}}{\ln (2)}$

Step 10.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 \ln^{2} (2) - 4 \ln (2) + 2}{\ln^{2} (2)} \cdot \frac{1}{\ln (2)}$

Step 10.3

Multiply

$\frac{2 \ln^{2} (2) - 4 \ln (2) + 2}{\ln^{2} (2)} \cdot \frac{1}{\ln (2)}$ .

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Step 10.3.1

Multiply

$\frac{2 \ln^{2} (2) - 4 \ln (2) + 2}{\ln^{2} (2)}$ by

$\frac{1}{\ln (2)}$ .

$\frac{2 \ln^{2} (2) - 4 \ln (2) + 2}{\ln^{2} (2) \ln (2)}$

Step 10.3.2

Multiply

$\ln^{2} (2)$ by

$\ln (2)$ by adding the exponents.

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Step 10.3.2.1

Multiply

$\ln^{2} (2)$ by

$\ln (2)$ .

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Step 10.3.2.1.1

Raise

$\ln (2)$ to the power of

$1$ .

$\frac{2 \ln^{2} (2) - 4 \ln (2) + 2}{\ln^{2} (2) \ln^{1} (2)}$

Step 10.3.2.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 \ln^{2} (2) - 4 \ln (2) + 2}{{\ln (2)}^{2 + 1}}$

Step 10.3.2.2

Add

$2$ and

$1$ .

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

Step 11

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.56547557 \dots$

Step 12