Calculus Examples

$\int \frac{1}{{(1 + x^{2})}^{\frac{3}{2}}} d x$

Step 1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\int \frac{1}{\sqrt{{(1 + x^{2})}^{3}}} d x$

Step 2

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.

$\int \frac{1}{\sqrt{{(1 + \tan^{2} (t))}^{3}}} \sec^{2} (t) d t$

Step 3

Simplify terms.

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Step 3.1

Simplify $\sqrt{{(1 + \tan^{2} (t))}^{3}}$ .

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Step 3.1.1

Rearrange terms.

$\int \frac{1}{\sqrt{{(\tan^{2} (t) + 1)}^{3}}} \sec^{2} (t) d t$

Step 3.1.2

Apply pythagorean identity.

$\int \frac{1}{\sqrt{{(\sec^{2} (t))}^{3}}} \sec^{2} (t) d t$

Step 3.1.3

Multiply the exponents in ${(\sec^{2} (t))}^{3}$ .

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Step 3.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int \frac{1}{\sqrt{{\sec (t)}^{2 \cdot 3}}} \sec^{2} (t) d t$

Step 3.1.3.2

Multiply $2$ by $3$ .

$\int \frac{1}{\sqrt{\sec^{6} (t)}} \sec^{2} (t) d t$

Step 3.1.4

Rewrite $\sec^{6} (t)$ as ${(\sec^{3} (t))}^{2}$ .

$\int \frac{1}{\sqrt{{(\sec^{3} (t))}^{2}}} \sec^{2} (t) d t$

Step 3.1.5

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

$\int \frac{1}{\sec^{3} (t)} \sec^{2} (t) d t$

Step 3.2

Reduce the expression by cancelling the common factors.

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Step 3.2.1

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

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Step 3.2.1.1

Factor $\sec^{2} (t)$ out of $\sec^{3} (t)$ .

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

Step 3.2.1.2

Cancel the common factor.

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

Step 3.2.1.3

Rewrite the expression.

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

Step 3.2.2

Simplify.

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Step 3.2.2.1

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

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

Step 3.2.2.2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (t)}$ .

$\int 1 \cos (t) d t$

Step 3.2.2.3

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

$\int \cos (t) d t$

Step 4

The integral of $\cos (t)$ with respect to $t$ is $\sin (t)$ .

$\sin (t) + C$

Step 5

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

$\sin (\arctan (x)) + C$