Calculus Examples

$\int {(36 - 9 x^{2})}^{- \frac{3}{2}} d x$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\int \frac{1}{{(36 - 9 x^{2})}^{\frac{3}{2}}} d x$

Step 2

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

$\int \frac{1}{\sqrt{{(36 - 9 x^{2})}^{3}}} d x$

Step 3

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

$\int \frac{1}{\sqrt{{(36 - 9 {(2 \sin (t))}^{2})}^{3}}} (2 \cos (t)) d t$

Step 4

Simplify terms.

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Step 4.1

Simplify $\sqrt{{(36 - 9 {(2 \sin (t))}^{2})}^{3}}$ .

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Apply the product rule to $2 \sin (t)$ .

$\int \frac{1}{\sqrt{{(36 - 9 (2^{2} \sin^{2} (t)))}^{3}}} (2 \cos (t)) d t$

Step 4.1.1.2

Raise $2$ to the power of $2$ .

$\int \frac{1}{\sqrt{{(36 - 9 (4 \sin^{2} (t)))}^{3}}} (2 \cos (t)) d t$

Step 4.1.1.3

Multiply $4$ by $- 9$ .

$\int \frac{1}{\sqrt{{(36 - 36 \sin^{2} (t))}^{3}}} (2 \cos (t)) d t$

Step 4.1.2

Factor $36$ out of $36$ .

$\int \frac{1}{\sqrt{{(36 (1) - 36 \sin^{2} (t))}^{3}}} (2 \cos (t)) d t$

Step 4.1.3

Factor $36$ out of $- 36 \sin^{2} (t)$ .

$\int \frac{1}{\sqrt{{(36 (1) + 36 (- \sin^{2} (t)))}^{3}}} (2 \cos (t)) d t$

Step 4.1.4

Factor $36$ out of $36 (1) + 36 (- \sin^{2} (t))$ .

$\int \frac{1}{\sqrt{{(36 (1 - \sin^{2} (t)))}^{3}}} (2 \cos (t)) d t$

Step 4.1.5

Apply pythagorean identity.

$\int \frac{1}{\sqrt{{(36 \cos^{2} (t))}^{3}}} (2 \cos (t)) d t$

Step 4.1.6

Apply the product rule to $36 \cos^{2} (t)$ .

$\int \frac{1}{\sqrt{36^{3} {(\cos^{2} (t))}^{3}}} (2 \cos (t)) d t$

Step 4.1.7

Raise $36$ to the power of $3$ .

$\int \frac{1}{\sqrt{46656 {(\cos^{2} (t))}^{3}}} (2 \cos (t)) d t$

Step 4.1.8

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

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Step 4.1.8.1

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

$\int \frac{1}{\sqrt{46656 {\cos (t)}^{2 \cdot 3}}} (2 \cos (t)) d t$

Step 4.1.8.2

Multiply $2$ by $3$ .

$\int \frac{1}{\sqrt{46656 \cos^{6} (t)}} (2 \cos (t)) d t$

Step 4.1.9

Rewrite $46656 \cos^{6} (t)$ as ${(216 \cos^{3} (t))}^{2}$ .

$\int \frac{1}{\sqrt{{(216 \cos^{3} (t))}^{2}}} (2 \cos (t)) d t$

Step 4.1.10

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

$\int \frac{1}{216 \cos^{3} (t)} (2 \cos (t)) d t$

Step 4.2

Cancel the common factor of $2 \cos (t)$ .

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Step 4.2.1

Factor $2 \cos (t)$ out of $216 \cos^{3} (t)$ .

$\int \frac{1}{2 \cos (t) (108 \cos^{2} (t))} (2 \cos (t)) d t$

Step 4.2.2

Cancel the common factor.

$\int \frac{1}{2 \cos (t) (108 \cos^{2} (t))} (2 \cos (t)) d t$

Step 4.2.3

Rewrite the expression.

$\int \frac{1}{108 \cos^{2} (t)} d t$

Step 5

Since $\frac{1}{108}$ is constant with respect to $t$ , move $\frac{1}{108}$ out of the integral.

$\frac{1}{108} \int \frac{1}{\cos^{2} (t)} d t$

Step 6

Convert from $\frac{1}{\cos^{2} (t)}$ to $\sec^{2} (t)$ .

$\frac{1}{108} \int \sec^{2} (t) d t$

Step 7

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

$\frac{1}{108} (\tan (t) + C)$

Step 8

Simplify.

$\frac{1}{108} \tan (t) + C$

Step 9

Replace all occurrences of $t$ with $\arcsin (\frac{x}{2})$ .

$\frac{1}{108} \tan (\arcsin (\frac{x}{2})) + C$

Step 10

Reorder terms.

$\frac{1}{108} \tan (\arcsin (\frac{1}{2} x)) + C$