Calculus Examples

$\int \frac{x^{2}}{{(4 - x^{2})}^{\frac{3}{2}}} d x$

Step 1

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

$\int \frac{x^{2}}{\sqrt{{(4 - x^{2})}^{3}}} d x$

Step 2

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.

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

Step 3

Simplify terms.

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Step 3.1

Simplify $\sqrt{{(4 - {(2 \sin (t))}^{2})}^{3}}$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

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

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

Step 3.1.1.2

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

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

Step 3.1.1.3

Multiply $4$ by $- 1$ .

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

Step 3.1.2

Factor $4$ out of $4$ .

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

Step 3.1.3

Factor $4$ out of $- 4 \sin^{2} (t)$ .

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

Step 3.1.4

Factor $4$ out of $4 (1) + 4 (- \sin^{2} (t))$ .

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

Step 3.1.5

Apply pythagorean identity.

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

Step 3.1.6

Apply the product rule to $4 \cos^{2} (t)$ .

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

Step 3.1.7

Raise $4$ to the power of $3$ .

$\int \frac{{(2 \sin (t))}^{2}}{\sqrt{64 {(\cos^{2} (t))}^{3}}} (2 \cos (t)) d t$

Step 3.1.8

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

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Step 3.1.8.1

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

$\int \frac{{(2 \sin (t))}^{2}}{\sqrt{64 {\cos (t)}^{2 \cdot 3}}} (2 \cos (t)) d t$

Step 3.1.8.2

Multiply $2$ by $3$ .

$\int \frac{{(2 \sin (t))}^{2}}{\sqrt{64 \cos^{6} (t)}} (2 \cos (t)) d t$

Step 3.1.9

Rewrite $64 \cos^{6} (t)$ as ${(8 \cos^{3} (t))}^{2}$ .

$\int \frac{{(2 \sin (t))}^{2}}{\sqrt{{(8 \cos^{3} (t))}^{2}}} (2 \cos (t)) d t$

Step 3.1.10

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

$\int \frac{{(2 \sin (t))}^{2}}{8 \cos^{3} (t)} (2 \cos (t)) d t$

Step 3.2

Reduce the expression by cancelling the common factors.

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Step 3.2.1

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

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Step 3.2.1.1

Factor $2 \cos (t)$ out of $8 \cos^{3} (t)$ .

$\int \frac{{(2 \sin (t))}^{2}}{2 \cos (t) (4 \cos^{2} (t))} (2 \cos (t)) d t$

Step 3.2.1.2

Cancel the common factor.

$\int \frac{{(2 \sin (t))}^{2}}{2 \cos (t) (4 \cos^{2} (t))} (2 \cos (t)) d t$

Step 3.2.1.3

Rewrite the expression.

$\int \frac{{(2 \sin (t))}^{2}}{4 \cos^{2} (t)} d t$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Factor $2$ out of $2 \sin (t)$ .

$\int \frac{{(2 (\sin (t)))}^{2}}{4 \cos^{2} (t)} d t$

Step 3.2.2.2

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

$\int \frac{2^{2} \sin^{2} (t)}{4 \cos^{2} (t)} d t$

Step 3.2.2.3

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

$\int \frac{4 \sin^{2} (t)}{4 \cos^{2} (t)} d t$

Step 3.2.2.4

Cancel the common factor of $4$ .

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Step 3.2.2.4.1

Cancel the common factor.

$\int \frac{4 \sin^{2} (t)}{4 \cos^{2} (t)} d t$

Step 3.2.2.4.2

Rewrite the expression.

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

Step 3.2.2.5

Convert from $\frac{\sin^{2} (t)}{\cos^{2} (t)}$ to $\tan^{2} (t)$ .

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

Step 4

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

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

Apply the constant rule.

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

Step 7

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

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

Step 8

Simplify.

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

Step 9

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

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

Step 10

Reorder terms.

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