Calculus Examples

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

Step 1

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.

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

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

Rearrange terms.

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

Step 2.1.2

Apply pythagorean identity.

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

Step 2.1.3

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

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

Step 2.2

Simplify.

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Step 2.2.1

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

$\int \frac{\sec (t)}{\frac{\sin (t)}{\cos (t)}} \sec^{2} (t) d t$

Step 2.2.2

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

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

Step 2.2.3

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (t)}{\cos (t)}$ .

$\int \frac{1}{\cos (t)} \cdot \frac{\cos (t)}{\sin (t)} \sec^{2} (t) d t$

Step 2.2.4

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

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Step 2.2.4.1

Cancel the common factor.

$\int \frac{1}{\cos (t)} \cdot \frac{\cos (t)}{\sin (t)} \sec^{2} (t) d t$

Step 2.2.4.2

Rewrite the expression.

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

Step 2.2.5

Convert from $\frac{1}{\sin (t)}$ to $\csc (t)$ .

$\int \csc (t) \sec^{2} (t) d t$

Step 3

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

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

Step 4

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

$\int \csc^{1} (t) (1 + \tan^{2} (t)) d t$

Step 5

Simplify terms.

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Step 5.1

Apply the distributive property.

$\int \csc^{1} (t) \cdot 1 + \csc^{1} (t) \tan^{2} (t) d t$

Step 5.2

Simplify each term.

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

The integral of $\csc (t)$ with respect to $t$ is $\ln (| \csc (t) - \cot (t) |)$ .

$\ln (| \csc (t) - \cot (t) |) + C + \int \csc (t) \tan^{2} (t) d t$

Step 8

Apply the reciprocal identity to $\csc (t)$ .

$\ln (| \csc (t) - \cot (t) |) + C + \int \frac{1}{\sin (t)} \tan^{2} (t) d t$

Step 9

Write $\tan (t)$ in sines and cosines using the quotient identity.

$\ln (| \csc (t) - \cot (t) |) + C + \int \frac{1}{\sin (t)} {(\frac{\sin (t)}{\cos (t)})}^{2} d t$

Step 10

Simplify.

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Step 10.1

Apply the product rule to $\frac{\sin (t)}{\cos (t)}$ .

$\ln (| \csc (t) - \cot (t) |) + C + \int \frac{1}{\sin (t)} \cdot \frac{{\sin (t)}^{2}}{{\cos (t)}^{2}} d t$

Step 10.2

Combine.

$\ln (| \csc (t) - \cot (t) |) + C + \int \frac{1 {\sin (t)}^{2}}{\sin (t) {\cos (t)}^{2}} d t$

Step 10.3

Cancel the common factor of ${\sin (t)}^{2}$ and $\sin (t)$ .

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Step 10.3.1

Factor $\sin (t)$ out of $1 {\sin (t)}^{2}$ .

$\ln (| \csc (t) - \cot (t) |) + C + \int \frac{\sin (t) (1 \sin (t))}{\sin (t) {\cos (t)}^{2}} d t$

Step 10.3.2

Cancel the common factors.

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Step 10.3.2.1

Factor $\sin (t)$ out of $\sin (t) {\cos (t)}^{2}$ .

$\ln (| \csc (t) - \cot (t) |) + C + \int \frac{\sin (t) (1 \sin (t))}{\sin (t) ({\cos (t)}^{2})} d t$

Step 10.3.2.2

Cancel the common factor.

$\ln (| \csc (t) - \cot (t) |) + C + \int \frac{\sin (t) (1 \sin (t))}{\sin (t) {\cos (t)}^{2}} d t$

Step 10.3.2.3

Rewrite the expression.

$\ln (| \csc (t) - \cot (t) |) + C + \int \frac{1 \sin (t)}{{\cos (t)}^{2}} d t$

Step 10.4

Multiply $\sin (t)$ by $1$ .

$\ln (| \csc (t) - \cot (t) |) + C + \int \frac{\sin (t)}{\cos^{2} (t)} d t$

Step 11

Multiply by $1$ .

$\ln (| \csc (t) - \cot (t) |) + C + \int \frac{\sin (t) \cdot 1}{\cos^{2} (t)} d t$

Step 12

Factor $\cos (t)$ out of $\cos^{2} (t)$ .

$\ln (| \csc (t) - \cot (t) |) + C + \int \frac{\sin (t) \cdot 1}{\cos (t) \cos (t)} d t$

Step 13

Separate fractions.

$\ln (| \csc (t) - \cot (t) |) + C + \int \frac{\sin (t)}{\cos (t)} \cdot \frac{1}{\cos (t)} d t$

Step 14

Convert from $\frac{\sin (t)}{\cos (t)}$ to $\tan (t)$ .

$\ln (| \csc (t) - \cot (t) |) + C + \int \tan (t) \frac{1}{\cos (t)} d t$

Step 15

Convert from $\frac{1}{\cos (t)}$ to $\sec (t)$ .

$\ln (| \csc (t) - \cot (t) |) + C + \int \tan (t) \sec (t) d t$

Step 16

Since the derivative of $\sec (t)$ is $\tan (t) \sec (t)$ , the integral of $\tan (t) \sec (t)$ is $\sec (t)$ .

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

Step 17

Simplify.

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

Step 18

Replace all occurrences of $t$ with $\arctan (x)$ .

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