Calculus Examples

\ln (x + \sqrt{x^{2} - 1})

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

Step 1

Find the

y

$y$ value at

x = 1

$x = 1$ .

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Step 1.1

Replace the variable

x

$x$ with

1

$1$ in the expression.

f (1) = \ln ((1) + \sqrt{((1) + 1) ((1) - 1)})

$f (1) = \ln ((1) + \sqrt{((1) + 1) ((1) - 1)})$

Step 1.2

Simplify the result.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Add

1

$1$ and

1

$1$ .

f (1) = \ln (1 + \sqrt{2 (1 - 1)})

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

Step 1.2.1.2

Subtract

1

$1$ from

1

$1$ .

f (1) = \ln (1 + \sqrt{2 \cdot 0})

$f (1) = \ln (1 + \sqrt{2 \cdot 0})$

Step 1.2.1.3

Multiply

2

$2$ by

0

$0$ .

f (1) = \ln (1 + \sqrt{0})

$f (1) = \ln (1 + \sqrt{0})$

Step 1.2.1.4

Rewrite

0

$0$ as

0^{2}

$0^{2}$ .

f (1) = \ln (1 + \sqrt{0^{2}})

$f (1) = \ln (1 + \sqrt{0^{2}})$

Step 1.2.1.5

Pull terms out from under the radical, assuming positive real numbers.

f (1) = \ln (1 + 0)

$f (1) = \ln (1 + 0)$

f (1) = \ln (1 + 0)

$f (1) = \ln (1 + 0)$

Step 1.2.2

Add

1

$1$ and

0

$0$ .

f (1) = \ln (1)

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

Step 1.2.3

The natural logarithm of

1

$1$ is

0

$0$ .

f (1) = 0

$f (1) = 0$

Step 1.2.4

The final answer is

0

$0$ .

0

$0$

0

$0$

Step 1.3

The

y

$y$ value at

x = 1

$x = 1$ is

0

$0$ .

y = 0

$y = 0$

y = 0

$y = 0$

Step 2

Find the

y

$y$ value at

x = 2

$x = 2$ .

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Step 2.1

Replace the variable

x

$x$ with

2

$2$ in the expression.

f (2) = \ln ((2) + \sqrt{((2) + 1) ((2) - 1)})

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

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Add

2

$2$ and

1

$1$ .

f (2) = \ln (2 + \sqrt{3 (2 - 1)})

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

Step 2.2.1.2

Subtract

1

$1$ from

2

$2$ .

f (2) = \ln (2 + \sqrt{3 \cdot 1})

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

Step 2.2.1.3

Multiply

3

$3$ by

1

$1$ .

f (2) = \ln (2 + \sqrt{3})

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

f (2) = \ln (2 + \sqrt{3})

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

Step 2.2.2

The final answer is

\ln (2 + \sqrt{3})

$\ln (2 + \sqrt{3})$ .

\ln (2 + \sqrt{3})

$\ln (2 + \sqrt{3})$

\ln (2 + \sqrt{3})

$\ln (2 + \sqrt{3})$

Step 2.3

The

y

$y$ value at

x = 2

$x = 2$ is

\ln (2 + \sqrt{3})

$\ln (2 + \sqrt{3})$ .

y = \ln (2 + \sqrt{3})

$y = \ln (2 + \sqrt{3})$

y = \ln (2 + \sqrt{3})

$y = \ln (2 + \sqrt{3})$

Step 3

Find the

y

$y$ value at

x = 3

$x = 3$ .

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Step 3.1

Replace the variable

x

$x$ with

3

$3$ in the expression.

f (3) = \ln ((3) + \sqrt{((3) + 1) ((3) - 1)})

$f (3) = \ln ((3) + \sqrt{((3) + 1) ((3) - 1)})$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Add

3

$3$ and

1

$1$ .

f (3) = \ln (3 + \sqrt{4 (3 - 1)})

$f (3) = \ln (3 + \sqrt{4 (3 - 1)})$

Step 3.2.1.2

Subtract

1

$1$ from

3

$3$ .

f (3) = \ln (3 + \sqrt{4 \cdot 2})

$f (3) = \ln (3 + \sqrt{4 \cdot 2})$

Step 3.2.1.3

Multiply

4

$4$ by

2

$2$ .

f (3) = \ln (3 + \sqrt{8})

$f (3) = \ln (3 + \sqrt{8})$

Step 3.2.1.4

Rewrite

8

$8$ as

2^{2} \cdot 2

$2^{2} \cdot 2$ .

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Step 3.2.1.4.1

Factor

4

$4$ out of

8

$8$ .

f (3) = \ln (3 + \sqrt{4 (2)})

$f (3) = \ln (3 + \sqrt{4 (2)})$

Step 3.2.1.4.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

f (3) = \ln (3 + \sqrt{2^{2} \cdot 2})

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

f (3) = \ln (3 + \sqrt{2^{2} \cdot 2})

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

Step 3.2.1.5

Pull terms out from under the radical.

f (3) = \ln (3 + 2 \sqrt{2})

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

f (3) = \ln (3 + 2 \sqrt{2})

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

Step 3.2.2

The final answer is

\ln (3 + 2 \sqrt{2})

$\ln (3 + 2 \sqrt{2})$ .

\ln (3 + 2 \sqrt{2})

$\ln (3 + 2 \sqrt{2})$

\ln (3 + 2 \sqrt{2})

$\ln (3 + 2 \sqrt{2})$

Step 3.3

The

y

$y$ value at

x = 3

$x = 3$ is

\ln (3 + 2 \sqrt{2})

$\ln (3 + 2 \sqrt{2})$ .

y = \ln (3 + 2 \sqrt{2})

$y = \ln (3 + 2 \sqrt{2})$

y = \ln (3 + 2 \sqrt{2})

$y = \ln (3 + 2 \sqrt{2})$

Step 4

List the points to graph.

(1, 0), (2, 1.31695789), (3, 1.76274717)

$(1, 0), (2, 1.31695789), (3, 1.76274717)$

Step 5

Select a few points to graph.

\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 1.317 \\ 3 & 1.763 \end{array}

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 1.317 \\ 3 & 1.763 \end{array}$

Step 6