Calculus Examples

$y = 3 x^{2} \ln (x)$ , $y = 12 \ln (x)$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Eliminate the equal sides of each equation and combine.

$x^{2} \ln (x^{3}) = \ln (x^{12})$

Step 1.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = 1, 2$

Step 1.3

Evaluate $y$ when $x = 1$ .

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Step 1.3.1

Substitute $1$ for $x$ .

$y = \ln ({(1)}^{12})$

Step 1.3.2

Substitute $1$ for $x$ in $y = \ln ({(1)}^{12})$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = \ln (1^{12})$

Step 1.3.2.2

Remove parentheses.

$y = \ln ({(1)}^{12})$

Step 1.3.2.3

Simplify $\ln ({(1)}^{12})$ .

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Step 1.3.2.3.1

One to any power is one.

$y = \ln (1)$

Step 1.3.2.3.2

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

$y = 0$

Step 1.4

Evaluate $y$ when $x = 2$ .

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Step 1.4.1

Substitute $2$ for $x$ .

$y = \ln ({(2)}^{12})$

Step 1.4.2

Substitute $2$ for $x$ in $y = \ln ({(2)}^{12})$ and solve for $y$ .

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Step 1.4.2.1

Remove parentheses.

$y = \ln (2^{12})$

Step 1.4.2.2

Remove parentheses.

$y = \ln ({(2)}^{12})$

Step 1.4.2.3

Raise $2$ to the power of $12$ .

$y = \ln (4096)$

Step 1.5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(1, 0)$

$(2, \ln (4096))$

$(1, 0)$

$(2, \ln (4096))$

Step 2

The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.

$A r e a = \int_{1}^{2} \ln (x^{12}) d x - \int_{1}^{2} x^{2} \ln (x^{3}) d x$

Step 3

Integrate to find the area between $1$ and $2$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{1}^{2} \ln (x^{12}) - (x^{2} \ln (x^{3})) d x$

Step 3.2

Remove parentheses.

$\int_{1}^{2} \ln (x^{12}) - x^{2} \ln (x^{3}) d x$

Step 3.3

Rewrite $\ln (x^{12}) - x^{2} \ln (x^{3})$ as $12 \ln (x) - x^{2} (3 \ln (x))$ .

$\int_{1}^{2} 12 \ln (x) - x^{2} (3 \ln (x)) d x$

Step 3.4

Split the single integral into multiple integrals.

$\int_{1}^{2} 12 \ln (x) d x + \int_{1}^{2} - x^{2} (3 \ln (x)) d x$

Step 3.5

Multiply $3$ by $- 1$ .

$\int_{1}^{2} 12 \ln (x) d x + \int_{1}^{2} - 3 x^{2} \ln (x) d x$

Step 3.6

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$12 \int_{1}^{2} \ln (x) d x + \int_{1}^{2} - 3 x^{2} \ln (x) d x$

Step 3.7

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

$12 (\ln (x) x]_{1}^{2} - \int_{1}^{2} x \frac{1}{x} d x) + \int_{1}^{2} - 3 x^{2} \ln (x) d x$

Step 3.8

Simplify.

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Step 3.8.1

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

$12 (\ln (x) x]_{1}^{2} - \int_{1}^{2} \frac{x}{x} d x) + \int_{1}^{2} - 3 x^{2} \ln (x) d x$

Step 3.8.2

Cancel the common factor of $x$ .

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Step 3.8.2.1

Cancel the common factor.

$12 (\ln (x) x]_{1}^{2} - \int_{1}^{2} \frac{x}{x} d x) + \int_{1}^{2} - 3 x^{2} \ln (x) d x$

Step 3.8.2.2

Rewrite the expression.

$12 (\ln (x) x]_{1}^{2} - \int_{1}^{2} d x) + \int_{1}^{2} - 3 x^{2} \ln (x) d x$

Step 3.9

Apply the constant rule.

$12 (\ln (x) x]_{1}^{2} - (x]_{1}^{2})) + \int_{1}^{2} - 3 x^{2} \ln (x) d x$

Step 3.10

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$12 (\ln (x) x]_{1}^{2} - (x]_{1}^{2})) - 3 \int_{1}^{2} x^{2} \ln (x) d x$

Step 3.11

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

$12 (\ln (x) x]_{1}^{2} - (x]_{1}^{2})) - 3 (\ln (x) (\frac{1}{3} x^{3})]_{1}^{2} - \int_{1}^{2} \frac{1}{3} x^{3} \frac{1}{x} d x)$

Step 3.12

Simplify.

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Step 3.12.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$12 (\ln (x) x]_{1}^{2} - (x]_{1}^{2})) - 3 (\ln (x) \frac{x^{3}}{3}]_{1}^{2} - \int_{1}^{2} \frac{1}{3} x^{3} \frac{1}{x} d x)$

Step 3.12.2

Combine $\ln (x)$ and $\frac{x^{3}}{3}$ .

$12 (\ln (x) x]_{1}^{2} - (x]_{1}^{2})) - 3 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \int_{1}^{2} \frac{1}{3} x^{3} \frac{1}{x} d x)$

Step 3.12.3

Combine $\frac{1}{3}$ and $x^{3}$ .

$12 (\ln (x) x]_{1}^{2} - (x]_{1}^{2})) - 3 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \int_{1}^{2} \frac{x^{3}}{3} \cdot \frac{1}{x} d x)$

Step 3.12.4

Multiply $\frac{x^{3}}{3}$ by $\frac{1}{x}$ .

$12 (\ln (x) x]_{1}^{2} - (x]_{1}^{2})) - 3 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \int_{1}^{2} \frac{x^{3}}{3 x} d x)$

Step 3.12.5

Cancel the common factor of $x^{3}$ and $x$ .

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Step 3.12.5.1

Factor $x$ out of $x^{3}$ .

$12 (\ln (x) x]_{1}^{2} - (x]_{1}^{2})) - 3 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \int_{1}^{2} \frac{x \cdot x^{2}}{3 x} d x)$

Step 3.12.5.2

Cancel the common factors.

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Step 3.12.5.2.1

Factor $x$ out of $3 x$ .

$12 (\ln (x) x]_{1}^{2} - (x]_{1}^{2})) - 3 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \int_{1}^{2} \frac{x \cdot x^{2}}{x \cdot 3} d x)$

Step 3.12.5.2.2

Cancel the common factor.

$12 (\ln (x) x]_{1}^{2} - (x]_{1}^{2})) - 3 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \int_{1}^{2} \frac{x \cdot x^{2}}{x \cdot 3} d x)$

Step 3.12.5.2.3

Rewrite the expression.

$12 (\ln (x) x]_{1}^{2} - (x]_{1}^{2})) - 3 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \int_{1}^{2} \frac{x^{2}}{3} d x)$

Step 3.13

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$12 (\ln (x) x]_{1}^{2} - (x]_{1}^{2})) - 3 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - (\frac{1}{3} \int_{1}^{2} x^{2} d x))$

Step 3.14

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

$12 (\ln (x) x]_{1}^{2} - (x]_{1}^{2})) - 3 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{1}{3} \frac{1}{3} x^{3}]_{1}^{2})$

Step 3.15

Simplify the answer.

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Step 3.15.1

Simplify.

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Step 3.15.1.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$12 (\ln (x) x]_{1}^{2} - (x]_{1}^{2})) - 3 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{1}{3} \frac{x^{3}}{3}]_{1}^{2})$

Step 3.15.1.2

Combine $\frac{x^{3}}{3}]_{1}^{2}$ and $\frac{1}{3}$ .

$12 (\ln (x) x]_{1}^{2} - (x]_{1}^{2})) - 3 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3})$

Step 3.15.2

Substitute and simplify.

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Step 3.15.2.1

Evaluate $\ln (x) x$ at $2$ and at $1$ .

$12 ((\ln (2) \cdot 2) - \ln (1) \cdot 1 - (x]_{1}^{2})) - 3 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3})$

Step 3.15.2.2

Evaluate $x$ at $2$ and at $1$ .

$12 ((\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)) - 3 (\frac{\ln (x) x^{3}}{3}]_{1}^{2} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3})$

Step 3.15.2.3

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

$12 ((\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)) - 3 ((\frac{\ln (2) \cdot 2^{3}}{3}) - \frac{\ln (1) \cdot 1^{3}}{3} - \frac{\frac{x^{3}}{3}]_{1}^{2}}{3})$

Step 3.15.2.4

Evaluate $\frac{x^{3}}{3}$ at $2$ and at $1$ .

$12 ((\ln (2) \cdot 2) - \ln (1) \cdot 1 - ((2) - 1)) - 3 ((\frac{\ln (2) \cdot 2^{3}}{3}) - \frac{\ln (1) \cdot 1^{3}}{3} - \frac{(\frac{2^{3}}{3}) - \frac{1^{3}}{3}}{3})$

Step 3.15.2.5

Simplify.

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Step 3.15.2.5.1

Move $2$ to the left of $\ln (2)$ .

$12 (2 \cdot \ln (2) - \ln (1) \cdot 1 - ((2) - 1)) - 3 ((\frac{\ln (2) \cdot 2^{3}}{3}) - \frac{\ln (1) \cdot 1^{3}}{3} - \frac{(\frac{2^{3}}{3}) - \frac{1^{3}}{3}}{3})$

Step 3.15.2.5.2

Multiply $- 1$ by $1$ .

$12 (2 \ln (2) - \ln (1) - ((2) - 1)) - 3 ((\frac{\ln (2) \cdot 2^{3}}{3}) - \frac{\ln (1) \cdot 1^{3}}{3} - \frac{(\frac{2^{3}}{3}) - \frac{1^{3}}{3}}{3})$

Step 3.15.2.5.3

Subtract $1$ from $2$ .

$12 (2 \ln (2) - \ln (1) - 1 \cdot 1) - 3 ((\frac{\ln (2) \cdot 2^{3}}{3}) - \frac{\ln (1) \cdot 1^{3}}{3} - \frac{(\frac{2^{3}}{3}) - \frac{1^{3}}{3}}{3})$

Step 3.15.2.5.4

Multiply $- 1$ by $1$ .

$12 (2 \ln (2) - \ln (1) - 1) - 3 ((\frac{\ln (2) \cdot 2^{3}}{3}) - \frac{\ln (1) \cdot 1^{3}}{3} - \frac{(\frac{2^{3}}{3}) - \frac{1^{3}}{3}}{3})$

Step 3.15.2.5.5

Raise $2$ to the power of $3$ .

$12 (2 \ln (2) - \ln (1) - 1) - 3 (\frac{\ln (2) \cdot 8}{3} - \frac{\ln (1) \cdot 1^{3}}{3} - \frac{(\frac{2^{3}}{3}) - \frac{1^{3}}{3}}{3})$

Step 3.15.2.5.6

Move $8$ to the left of $\ln (2)$ .

$12 (2 \ln (2) - \ln (1) - 1) - 3 (\frac{8 \cdot \ln (2)}{3} - \frac{\ln (1) \cdot 1^{3}}{3} - \frac{(\frac{2^{3}}{3}) - \frac{1^{3}}{3}}{3})$

Step 3.15.2.5.7

One to any power is one.

$12 (2 \ln (2) - \ln (1) - 1) - 3 (\frac{8 \ln (2)}{3} - \frac{\ln (1) \cdot 1}{3} - \frac{(\frac{2^{3}}{3}) - \frac{1^{3}}{3}}{3})$

Step 3.15.2.5.8

Multiply $\ln (1)$ by $1$ .

$12 (2 \ln (2) - \ln (1) - 1) - 3 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{(\frac{2^{3}}{3}) - \frac{1^{3}}{3}}{3})$

Step 3.15.2.5.9

Raise $2$ to the power of $3$ .

$12 (2 \ln (2) - \ln (1) - 1) - 3 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{\frac{8}{3} - \frac{1^{3}}{3}}{3})$

Step 3.15.2.5.10

One to any power is one.

$12 (2 \ln (2) - \ln (1) - 1) - 3 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{\frac{8}{3} - \frac{1}{3}}{3})$

Step 3.15.2.5.11

Combine the numerators over the common denominator.

$12 (2 \ln (2) - \ln (1) - 1) - 3 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{\frac{8 - 1}{3}}{3})$

Step 3.15.2.5.12

Subtract $1$ from $8$ .

$12 (2 \ln (2) - \ln (1) - 1) - 3 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{\frac{7}{3}}{3})$

Step 3.15.2.5.13

Rewrite $\frac{\frac{7}{3}}{3}$ as a product.

$12 (2 \ln (2) - \ln (1) - 1) - 3 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - (\frac{7}{3} \cdot \frac{1}{3}))$

Step 3.15.2.5.14

Multiply $\frac{7}{3}$ by $\frac{1}{3}$ .

$12 (2 \ln (2) - \ln (1) - 1) - 3 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{3 \cdot 3})$

Step 3.15.2.5.15

Multiply $3$ by $3$ .

$12 (2 \ln (2) - \ln (1) - 1) - 3 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9})$

Step 3.16

Simplify.

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Step 3.16.1

Simplify each term.

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Step 3.16.1.1

Simplify each term.

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Step 3.16.1.1.1

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

$12 (2 \ln (2) - 0 - 1) - 3 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9})$

Step 3.16.1.1.2

Multiply $- 1$ by $0$ .

$12 (2 \ln (2) + 0 - 1) - 3 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9})$

Step 3.16.1.2

Add $2 \ln (2)$ and $0$ .

$12 (2 \ln (2) - 1) - 3 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9})$

Step 3.16.1.3

Apply the distributive property.

$12 (2 \ln (2)) + 12 \cdot - 1 - 3 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9})$

Step 3.16.1.4

Multiply $2$ by $12$ .

$24 \ln (2) + 12 \cdot - 1 - 3 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9})$

Step 3.16.1.5

Multiply $12$ by $- 1$ .

$24 \ln (2) - 12 - 3 (\frac{8 \ln (2)}{3} - \frac{\ln (1)}{3} - \frac{7}{9})$

Step 3.16.1.6

Combine the numerators over the common denominator.

$24 \ln (2) - 12 - 3 (\frac{8 \ln (2) - \ln (1)}{3} + \frac{- 7}{9})$

Step 3.16.1.7

Simplify each term.

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Step 3.16.1.7.1

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

$24 \ln (2) - 12 - 3 (\frac{8 \ln (2) - 0}{3} + \frac{- 7}{9})$

Step 3.16.1.7.2

Multiply $- 1$ by $0$ .

$24 \ln (2) - 12 - 3 (\frac{8 \ln (2) + 0}{3} + \frac{- 7}{9})$

Step 3.16.1.8

Add $8 \ln (2)$ and $0$ .

$24 \ln (2) - 12 - 3 (\frac{8 \ln (2)}{3} + \frac{- 7}{9})$

Step 3.16.1.9

Move the negative in front of the fraction.

$24 \ln (2) - 12 - 3 (\frac{8 \ln (2)}{3} - \frac{7}{9})$

Step 3.16.1.10

Apply the distributive property.

$24 \ln (2) - 12 - 3 \frac{8 \ln (2)}{3} - 3 (- \frac{7}{9})$

Step 3.16.1.11

Cancel the common factor of $3$ .

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Step 3.16.1.11.1

Factor $3$ out of $- 3$ .

$24 \ln (2) - 12 + 3 (- 1) \frac{8 \ln (2)}{3} - 3 (- \frac{7}{9})$

Step 3.16.1.11.2

Cancel the common factor.

$24 \ln (2) - 12 + 3 \cdot - 1 \frac{8 \ln (2)}{3} - 3 (- \frac{7}{9})$

Step 3.16.1.11.3

Rewrite the expression.

$24 \ln (2) - 12 - (8 \ln (2)) - 3 (- \frac{7}{9})$

Step 3.16.1.12

Multiply $8$ by $- 1$ .

$24 \ln (2) - 12 - 8 \ln (2) - 3 (- \frac{7}{9})$

Step 3.16.1.13

Cancel the common factor of $3$ .

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Step 3.16.1.13.1

Move the leading negative in $- \frac{7}{9}$ into the numerator.

$24 \ln (2) - 12 - 8 \ln (2) - 3 (\frac{- 7}{9})$

Step 3.16.1.13.2

Factor $3$ out of $- 3$ .

$24 \ln (2) - 12 - 8 \ln (2) + 3 (- 1) \frac{- 7}{9}$

Step 3.16.1.13.3

Factor $3$ out of $9$ .

$24 \ln (2) - 12 - 8 \ln (2) + 3 \cdot - 1 \frac{- 7}{3 \cdot 3}$

Step 3.16.1.13.4

Cancel the common factor.

$24 \ln (2) - 12 - 8 \ln (2) + 3 \cdot - 1 \frac{- 7}{3 \cdot 3}$

Step 3.16.1.13.5

Rewrite the expression.

$24 \ln (2) - 12 - 8 \ln (2) - \frac{- 7}{3}$

Step 3.16.1.14

Simplify each term.

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Step 3.16.1.14.1

Move the negative in front of the fraction.

$24 \ln (2) - 12 - 8 \ln (2) - - \frac{7}{3}$

Step 3.16.1.14.2

Multiply $- - \frac{7}{3}$ .

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Step 3.16.1.14.2.1

Multiply $- 1$ by $- 1$ .

$24 \ln (2) - 12 - 8 \ln (2) + 1 (\frac{7}{3})$

Step 3.16.1.14.2.2

Multiply $\frac{7}{3}$ by $1$ .

$24 \ln (2) - 12 - 8 \ln (2) + \frac{7}{3}$

Step 3.16.2

Subtract $8 \ln (2)$ from $24 \ln (2)$ .

$16 \ln (2) - 12 + \frac{7}{3}$

Step 3.16.3

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$16 \ln (2) - 12 \cdot \frac{3}{3} + \frac{7}{3}$

Step 3.16.4

Combine $- 12$ and $\frac{3}{3}$ .

$16 \ln (2) + \frac{- 12 \cdot 3}{3} + \frac{7}{3}$

Step 3.16.5

Combine the numerators over the common denominator.

$16 \ln (2) + \frac{- 12 \cdot 3 + 7}{3}$

Step 3.16.6

Simplify the numerator.

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Step 3.16.6.1

Multiply $- 12$ by $3$ .

$16 \ln (2) + \frac{- 36 + 7}{3}$

Step 3.16.6.2

Add $- 36$ and $7$ .

$16 \ln (2) + \frac{- 29}{3}$

Step 3.16.7

Move the negative in front of the fraction.

$16 \ln (2) - \frac{29}{3}$

Step 4

Simplify each term.

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Step 4.1

Simplify $16 \ln (2)$ by moving $16$ inside the logarithm.

$\ln (2^{16}) - \frac{29}{3}$

Step 4.2

Raise $2$ to the power of $16$ .

$\ln (65536) - \frac{29}{3}$

Step 5