Calculus Examples

\frac{1}{x \sqrt{x^{2} + 1}}

$\frac{1}{x \sqrt{x^{2} + 1}}$

Step 1

Write

$\frac{1}{x \sqrt{x^{2} + 1}}$ as a function.

$f (x) = \frac{1}{x \sqrt{x^{2} + 1}}$

Step 2

The function

$F (x)$ can be found by finding the indefinite integral of the derivative

$f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{1}{x \sqrt{x^{2} + 1}} d x$

Step 4

Let

$x = \tan (t)$ , where

$- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then

$d x = \sec^{2} (t) d t$ . Note that since

$- \frac{π}{2} \leq t \leq \frac{π}{2}$ ,

$\sec^{2} (t)$ is positive.

$\int \frac{1}{\tan (t) \sqrt{\tan^{2} (t) + 1}} \sec^{2} (t) d t$

Step 5

Simplify terms.

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Step 5.1

Simplify

$\sqrt{\tan^{2} (t) + 1}$ .

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Step 5.1.1

Apply pythagorean identity.

$\int \frac{1}{\tan (t) \sqrt{\sec^{2} (t)}} \sec^{2} (t) d t$

Step 5.1.2

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

$\int \frac{1}{\tan (t) \sec (t)} \sec^{2} (t) d t$

Step 5.2

Simplify terms.

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Step 5.2.1

Cancel the common factor of

$\sec (t)$ .

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Step 5.2.1.1

Factor

$\sec (t)$ out of

$\tan (t) \sec (t)$ .

$\int \frac{1}{\sec (t) \tan (t)} \sec^{2} (t) d t$

Step 5.2.1.2

Factor

$\sec (t)$ out of

$\sec^{2} (t)$ .

$\int \frac{1}{\sec (t) \tan (t)} (\sec (t) \sec (t)) d t$

Step 5.2.1.3

Cancel the common factor.

$\int \frac{1}{\sec (t) \tan (t)} (\sec (t) \sec (t)) d t$

Step 5.2.1.4

Rewrite the expression.

$\int \frac{1}{\tan (t)} \sec (t) d t$

Step 5.2.2

Combine

$\frac{1}{\tan (t)}$ and

$\sec (t)$ .

$\int \frac{\sec (t)}{\tan (t)} d t$

Step 5.2.3

Simplify.

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Step 5.2.3.1

Rewrite

$\tan (t)$ in terms of sines and cosines.

$\int \frac{\sec (t)}{\frac{\sin (t)}{\cos (t)}} d t$

Step 5.2.3.2

Rewrite

$\sec (t)$ in terms of sines and cosines.

$\int \frac{\frac{1}{\cos (t)}}{\frac{\sin (t)}{\cos (t)}} d t$

Step 5.2.3.3

Multiply by the reciprocal of the fraction to divide by

$\frac{\sin (t)}{\cos (t)}$ .

$\int \frac{1}{\cos (t)} \cdot \frac{\cos (t)}{\sin (t)} d t$

Step 5.2.3.4

Cancel the common factor of

$\cos (t)$ .

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Step 5.2.3.4.1

Cancel the common factor.

$\int \frac{1}{\cos (t)} \cdot \frac{\cos (t)}{\sin (t)} d t$

Step 5.2.3.4.2

Rewrite the expression.

$\int \frac{1}{\sin (t)} d t$

Step 5.2.3.5

Convert from

$\frac{1}{\sin (t)}$ to

$\csc (t)$ .

$\int \csc (t) d t$

Step 6

The integral of

$\csc (t)$ with respect to

$t$ is

$\ln (| \csc (t) - \cot (t) |)$ .

$\ln (| \csc (t) - \cot (t) |) + C$

Step 7

Replace all occurrences of

$t$ with

$\arctan (x)$ .

$\ln (| \csc (\arctan (x)) - \cot (\arctan (x)) |) + C$

Step 8

The answer is the antiderivative of the function

$f (x) = \frac{1}{x \sqrt{x^{2} + 1}}$ .

$F (x) =$

$\ln (| \csc (\arctan (x)) - \cot (\arctan (x)) |) + C$