Calculus Examples

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

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.

Tap for more steps...

Step 5.1

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

Tap for more steps...

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.

Tap for more steps...

Step 5.2.1

Cancel the common factor of $\sec (t)$ .

Tap for more steps...

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.

Tap for more steps...

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 $\cos (t)$ .

Tap for more steps...

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

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