Calculus Examples

$\int_{2}^{8} \frac{1}{{(x^{2} - 1)}^{\frac{3}{2}}} d x$

Step 1

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

$\int_{2}^{8} \frac{1}{\sqrt{{(x^{2} - 1)}^{3}}} d x$

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.

$\int_{\frac{π}{3}}^{1.44546849} \frac{1}{\sqrt{{(\sec^{2} (t) - 1)}^{3}}} (\sec (t) \tan (t)) d t$

Step 3

Simplify terms.

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Step 3.1

Simplify $\sqrt{{(\sec^{2} (t) - 1)}^{3}}$ .

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Step 3.1.1

Apply pythagorean identity.

$\int_{\frac{π}{3}}^{1.44546849} \frac{1}{\sqrt{{(\tan^{2} (t))}^{3}}} (\sec (t) \tan (t)) d t$

Step 3.1.2

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

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Step 3.1.2.1

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

$\int_{\frac{π}{3}}^{1.44546849} \frac{1}{\sqrt{{\tan (t)}^{2 \cdot 3}}} (\sec (t) \tan (t)) d t$

Step 3.1.2.2

Multiply $2$ by $3$ .

$\int_{\frac{π}{3}}^{1.44546849} \frac{1}{\sqrt{\tan^{6} (t)}} (\sec (t) \tan (t)) d t$

Step 3.1.3

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

$\int_{\frac{π}{3}}^{1.44546849} \frac{1}{\sqrt{{(\tan^{3} (t))}^{2}}} (\sec (t) \tan (t)) d t$

Step 3.1.4

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

$\int_{\frac{π}{3}}^{1.44546849} \frac{1}{\tan^{3} (t)} (\sec (t) \tan (t)) d t$

Step 3.2

Simplify terms.

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Step 3.2.1

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

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Step 3.2.1.1

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

$\int_{\frac{π}{3}}^{1.44546849} \frac{1}{\tan (t) \tan^{2} (t)} (\sec (t) \tan (t)) d t$

Step 3.2.1.2

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

$\int_{\frac{π}{3}}^{1.44546849} \frac{1}{\tan (t) \tan^{2} (t)} (\tan (t) \sec (t)) d t$

Step 3.2.1.3

Cancel the common factor.

$\int_{\frac{π}{3}}^{1.44546849} \frac{1}{\tan (t) \tan^{2} (t)} (\tan (t) \sec (t)) d t$

Step 3.2.1.4

Rewrite the expression.

$\int_{\frac{π}{3}}^{1.44546849} \frac{1}{\tan^{2} (t)} \sec (t) d t$

Step 3.2.2

Combine $\frac{1}{\tan^{2} (t)}$ and $\sec (t)$ .

$\int_{\frac{π}{3}}^{1.44546849} \frac{\sec (t)}{\tan^{2} (t)} d t$

Step 4

Simplify.

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Step 4.1

Rewrite $\sec (x)$ as $\frac{\csc (x)}{\cot (x)}$ .

$\int_{\frac{π}{3}}^{1.44546849} \frac{\frac{\csc (t)}{\cot (t)}}{\tan^{2} (t)} d t$

Step 4.2

Apply the reciprocal identity.

$\int_{\frac{π}{3}}^{1.44546849} \frac{\frac{\csc (t)}{\cot (t)}}{{(\frac{1}{\cot (t)})}^{2}} d t$

Step 4.3

Simplify.

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Step 4.3.1

Multiply the numerator by the reciprocal of the denominator.

$\int_{\frac{π}{3}}^{1.44546849} \frac{\csc (t)}{\cot (t)} \cdot \frac{1}{{(\frac{1}{\cot (t)})}^{2}} d t$

Step 4.3.2

Combine.

$\int_{\frac{π}{3}}^{1.44546849} \frac{\csc (t) \cdot 1}{\cot (t) {(\frac{1}{\cot (t)})}^{2}} d t$

Step 4.3.3

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

$\int_{\frac{π}{3}}^{1.44546849} \frac{\csc (t)}{\cot (t) {(\frac{1}{\cot (t)})}^{2}} d t$

Step 4.3.4

Simplify the denominator.

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Step 4.3.4.1

Apply the product rule to $\frac{1}{\cot (t)}$ .

$\int_{\frac{π}{3}}^{1.44546849} \frac{\csc (t)}{\cot (t) \frac{1^{2}}{{\cot (t)}^{2}}} d t$

Step 4.3.4.2

One to any power is one.

$\int_{\frac{π}{3}}^{1.44546849} \frac{\csc (t)}{\cot (t) \frac{1}{{\cot (t)}^{2}}} d t$

Step 4.3.5

Combine $\cot (t)$ and $\frac{1}{{\cot (t)}^{2}}$ .

$\int_{\frac{π}{3}}^{1.44546849} \frac{\csc (t)}{\frac{\cot (t)}{{\cot (t)}^{2}}} d t$

Step 4.3.6

Reduce the expression $\frac{\cot (t)}{{\cot (t)}^{2}}$ by cancelling the common factors.

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Step 4.3.6.1

Multiply by $1$ .

$\int_{\frac{π}{3}}^{1.44546849} \frac{\csc (t)}{\frac{\cot (t) \cdot 1}{{\cot (t)}^{2}}} d t$

Step 4.3.6.2

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

$\int_{\frac{π}{3}}^{1.44546849} \frac{\csc (t)}{\frac{\cot (t) \cdot 1}{\cot (t) \cot (t)}} d t$

Step 4.3.6.3

Cancel the common factor.

$\int_{\frac{π}{3}}^{1.44546849} \frac{\csc (t)}{\frac{\cot (t) \cdot 1}{\cot (t) \cot (t)}} d t$

Step 4.3.6.4

Rewrite the expression.

$\int_{\frac{π}{3}}^{1.44546849} \frac{\csc (t)}{\frac{1}{\cot (t)}} d t$

Step 4.3.7

Multiply the numerator by the reciprocal of the denominator.

$\int_{\frac{π}{3}}^{1.44546849} \csc (t) \cot (t) d t$

Step 5

Since the derivative of $- \csc (t)$ is $\csc (t) \cot (t)$ , the integral of $\csc (t) \cot (t)$ is $- \csc (t)$ .

$- \csc (t)]_{\frac{π}{3}}^{1.44546849}$

Step 6

Evaluate $- \csc (t)$ at $1.44546849$ and at $\frac{π}{3}$ .

$- \csc (1.44546849) + \csc (\frac{π}{3})$

Step 7

The exact value of $\csc (\frac{π}{3})$ is $\frac{2}{\sqrt{3}}$ .

$- \csc (1.44546849) + \frac{2}{\sqrt{3}}$

Step 8

Simplify.

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Step 8.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$- \csc (1.44546849) + \frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 8.2

Combine and simplify the denominator.

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Step 8.2.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$- \csc (1.44546849) + \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 8.2.2

Raise $\sqrt{3}$ to the power of $1$ .

$- \csc (1.44546849) + \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 8.2.3

Raise $\sqrt{3}$ to the power of $1$ .

$- \csc (1.44546849) + \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 8.2.4

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

$- \csc (1.44546849) + \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 8.2.5

Add $1$ and $1$ .

$- \csc (1.44546849) + \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 8.2.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 8.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$- \csc (1.44546849) + \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 8.2.6.2

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

$- \csc (1.44546849) + \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 8.2.6.3

Combine $\frac{1}{2}$ and $2$ .

$- \csc (1.44546849) + \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 8.2.6.4

Cancel the common factor of $2$ .

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Step 8.2.6.4.1

Cancel the common factor.

$- \csc (1.44546849) + \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 8.2.6.4.2

Rewrite the expression.

$- \csc (1.44546849) + \frac{2 \sqrt{3}}{3^{1}}$

Step 8.2.6.5

Evaluate the exponent.

$- \csc (1.44546849) + \frac{2 \sqrt{3}}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \csc (1.44546849) + \frac{2 \sqrt{3}}{3}$

Decimal Form:

$0.14679527 \dots$

Step 10