Calculus Examples

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

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$1 + x^{2} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 1.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 1}$

Step 1.2.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

Step 1.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i$

Step 1.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i$

Step 1.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i, - i$

Step 1.3

The vertical asymptote occurs at $x = i, x = - i$ .

Vertical Asymptote: $x = i, x = - i$

Step 2

Find the point at $x = 1$ .

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Step 2.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \ln (1 + {(1)}^{2})$

Step 2.2

Simplify the result.

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Step 2.2.1

One to any power is one.

$f (1) = \ln (1 + 1)$

Step 2.2.2

Add $1$ and $1$ .

$f (1) = \ln (2)$

Step 2.2.3

The final answer is $\ln (2)$ .

$\ln (2)$

Step 2.3

Convert $\ln (2)$ to decimal.

$y = 0.69314718$

Step 3

Find the point at $x = 2$ .

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Step 3.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = \ln (1 + {(2)}^{2})$

Step 3.2

Simplify the result.

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Step 3.2.1

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

$f (2) = \ln (1 + 4)$

Step 3.2.2

Add $1$ and $4$ .

$f (2) = \ln (5)$

Step 3.2.3

The final answer is $\ln (5)$ .

$\ln (5)$

Step 3.3

Convert $\ln (5)$ to decimal.

$y = 1.60943791$

Step 4

Find the point at $x = 3$ .

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Step 4.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \ln (1 + {(3)}^{2})$

Step 4.2

Simplify the result.

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Step 4.2.1

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

$f (3) = \ln (1 + 9)$

Step 4.2.2

Add $1$ and $9$ .

$f (3) = \ln (10)$

Step 4.2.3

The final answer is $\ln (10)$ .

$\ln (10)$

Step 4.3

Convert $\ln (10)$ to decimal.

$y = 2.30258509$

Step 5

The log function can be graphed using the vertical asymptote at $x = i, x = - i$ and the points $(1, 0.69314718), (2, 1.60943791), (3, 2.30258509)$ .

Vertical Asymptote: $x = i, x = - i$

$\begin{array}{c} x & y \\ 1 & 0.693 \\ 2 & 1.609 \\ 3 & 2.303 \end{array}$

Step 6