Calculus Examples

$\int \frac{\sqrt{x^{2} - 1}}{x} d x$

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

Let $x = \sec (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \sec (t) \tan (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\sec (t) \tan (t)$ is positive.

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

Step 3

Simplify terms.

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Step 3.1

Simplify $\sqrt{\sec^{2} (t) - 1}$ .

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Step 3.1.1

Apply pythagorean identity.

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

Step 3.1.2

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

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

Step 3.2

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

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Step 3.2.1

Factor $\sec (t)$ out of $\sec (t) \tan (t)$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

$\int \tan (t) \tan (t) d t$

Step 4

Raise $\tan (t)$ to the power of $1$ .

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

Step 5

Raise $\tan (t)$ to the power of $1$ .

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

Step 6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int {\tan (t)}^{1 + 1} d t$

Step 7

Add $1$ and $1$ .

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

Step 8

Using the Pythagorean Identity, rewrite $\tan^{2} (t)$ as $- 1 + \sec^{2} (t)$ .

$\int - 1 + \sec^{2} (t) d t$

Step 9

Split the single integral into multiple integrals.

$\int - 1 d t + \int \sec^{2} (t) d t$

Step 10

Apply the constant rule.

$- t + C + \int \sec^{2} (t) d t$

Step 11

Since the derivative of $\tan (t)$ is $\sec^{2} (t)$ , the integral of $\sec^{2} (t)$ is $\tan (t)$ .

$- t + C + \tan (t) + C$

Step 12

Simplify.

$- t + \tan (t) + C$

Step 13

Replace all occurrences of $t$ with $arcsec (x)$ .

$- arcsec (x) + \tan (arcsec (x)) + C$