Calculus Examples

$\int \frac{x^{3}}{\sqrt{16 - x^{2}}} d x$

Step 1

Let $x = 4 \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = 4 \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $4 \cos (t)$ is positive.

$\int \frac{{(4 \sin (t))}^{3}}{\sqrt{16 - {(4 \sin (t))}^{2}}} (4 \cos (t)) d t$

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

Simplify $\sqrt{16 - {(4 \sin (t))}^{2}}$ .

Tap for more steps...

Step 2.1.1

Simplify each term.

Tap for more steps...

Step 2.1.1.1

Apply the product rule to $4 \sin (t)$ .

$\int \frac{{(4 \sin (t))}^{3}}{\sqrt{16 - (4^{2} \sin^{2} (t))}} (4 \cos (t)) d t$

Step 2.1.1.2

Raise $4$ to the power of $2$ .

$\int \frac{{(4 \sin (t))}^{3}}{\sqrt{16 - (16 \sin^{2} (t))}} (4 \cos (t)) d t$

Step 2.1.1.3

Multiply $16$ by $- 1$ .

$\int \frac{{(4 \sin (t))}^{3}}{\sqrt{16 - 16 \sin^{2} (t)}} (4 \cos (t)) d t$

Step 2.1.2

Factor $16$ out of $16$ .

$\int \frac{{(4 \sin (t))}^{3}}{\sqrt{16 (1) - 16 \sin^{2} (t)}} (4 \cos (t)) d t$

Step 2.1.3

Factor $16$ out of $- 16 \sin^{2} (t)$ .

$\int \frac{{(4 \sin (t))}^{3}}{\sqrt{16 (1) + 16 (- \sin^{2} (t))}} (4 \cos (t)) d t$

Step 2.1.4

Factor $16$ out of $16 (1) + 16 (- \sin^{2} (t))$ .

$\int \frac{{(4 \sin (t))}^{3}}{\sqrt{16 (1 - \sin^{2} (t))}} (4 \cos (t)) d t$

Step 2.1.5

Apply pythagorean identity.

$\int \frac{{(4 \sin (t))}^{3}}{\sqrt{16 \cos^{2} (t)}} (4 \cos (t)) d t$

Step 2.1.6

Rewrite $16 \cos^{2} (t)$ as ${(4 \cos (t))}^{2}$ .

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

Step 2.1.7

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

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

Step 2.2

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $4 \cos (t)$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

$\int {(4 \sin (t))}^{3} d t$

Step 2.2.2

Simplify.

Tap for more steps...

Step 2.2.2.1

Factor $4$ out of $4 \sin (t)$ .

$\int {(4 (\sin (t)))}^{3} d t$

Step 2.2.2.2

Apply the product rule to $4 (\sin (t))$ .

$\int 4^{3} \sin^{3} (t) d t$

Step 2.2.2.3

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

$\int 64 \sin^{3} (t) d t$

Step 3

Since $64$ is constant with respect to $t$ , move $64$ out of the integral.

$64 \int \sin^{3} (t) d t$

Step 4

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

$64 \int \sin^{2} (t) \sin (t) d t$

Step 5

Using the Pythagorean Identity, rewrite $\sin^{2} (t)$ as $1 - \cos^{2} (t)$ .

$64 \int (1 - \cos^{2} (t)) \sin (t) d t$

Step 6

Let $u = \cos (t)$ . Then $d u = - \sin (t) d t$ , so $- \frac{1}{\sin (t)} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 6.1

Let $u = \cos (t)$ . Find $\frac{d u}{d t}$ .

Tap for more steps...

Step 6.1.1

Differentiate $\cos (t)$ .

$\frac{d}{d t} [\cos (t)]$

Step 6.1.2

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

$- \sin (t)$

Step 6.2

Rewrite the problem using $u$ and $d u$ .

$64 \int - 1 + u^{2} d u$

Step 7

Split the single integral into multiple integrals.

$64 (\int - 1 d u + \int u^{2} d u)$

Step 8

Apply the constant rule.

$64 (- u + C + \int u^{2} d u)$

Step 9

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$64 (- u + C + \frac{1}{3} u^{3} + C)$

Step 10

Simplify.

Tap for more steps...

Step 10.1

Combine $\frac{1}{3}$ and $u^{3}$ .

$64 (- u + C + \frac{u^{3}}{3} + C)$

Step 10.2

Simplify.

$64 (- u + \frac{1}{3} u^{3}) + C$

Step 11

Substitute back in for each integration substitution variable.

Tap for more steps...

Step 11.1

Replace all occurrences of $u$ with $\cos (t)$ .

$64 (- \cos (t) + \frac{1}{3} \cos^{3} (t)) + C$

Step 11.2

Replace all occurrences of $t$ with $\arcsin (\frac{x}{4})$ .

$64 (- \cos (\arcsin (\frac{x}{4})) + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{4}))) + C$

Step 12

Simplify.

Tap for more steps...

Step 12.1

Simplify each term.

Tap for more steps...

Step 12.1.1

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{x}{4})}^{2}}, \frac{x}{4})$ , $(\sqrt{1^{2} - {(\frac{x}{4})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{x}{4})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{x}{4})}^{2}}, \frac{x}{4})$ . Therefore, $\cos (\arcsin (\frac{x}{4}))$ is $\sqrt{1 - {(\frac{x}{4})}^{2}}$ .

$64 (- \sqrt{1 - {(\frac{x}{4})}^{2}} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{4}))) + C$

Step 12.1.2

Rewrite $1$ as $1^{2}$ .

$64 (- \sqrt{1^{2} - {(\frac{x}{4})}^{2}} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{4}))) + C$

Step 12.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \frac{x}{4}$ .

$64 (- \sqrt{(1 + \frac{x}{4}) (1 - \frac{x}{4})} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{4}))) + C$

Step 12.1.4

Write $1$ as a fraction with a common denominator.

$64 (- \sqrt{(\frac{4}{4} + \frac{x}{4}) (1 - \frac{x}{4})} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{4}))) + C$

Step 12.1.5

Combine the numerators over the common denominator.

$64 (- \sqrt{\frac{4 + x}{4} (1 - \frac{x}{4})} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{4}))) + C$

Step 12.1.6

Write $1$ as a fraction with a common denominator.

$64 (- \sqrt{\frac{4 + x}{4} (\frac{4}{4} - \frac{x}{4})} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{4}))) + C$

Step 12.1.7

Combine the numerators over the common denominator.

$64 (- \sqrt{\frac{4 + x}{4} \cdot \frac{4 - x}{4}} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{4}))) + C$

Step 12.1.8

Multiply $\frac{4 + x}{4}$ by $\frac{4 - x}{4}$ .

$64 (- \sqrt{\frac{(4 + x) (4 - x)}{4 \cdot 4}} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{4}))) + C$

Step 12.1.9

Multiply $4$ by $4$ .

$64 (- \sqrt{\frac{(4 + x) (4 - x)}{16}} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{4}))) + C$

Step 12.1.10

Rewrite $\frac{(4 + x) (4 - x)}{16}$ as ${(\frac{1}{4})}^{2} ((4 + x) (4 - x))$ .

Tap for more steps...

Step 12.1.10.1

Factor the perfect power $1^{2}$ out of $(4 + x) (4 - x)$ .

$64 (- \sqrt{\frac{1^{2} ((4 + x) (4 - x))}{16}} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{4}))) + C$

Step 12.1.10.2

Factor the perfect power $4^{2}$ out of $16$ .

$64 (- \sqrt{\frac{1^{2} ((4 + x) (4 - x))}{4^{2} \cdot 1}} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{4}))) + C$

Step 12.1.10.3

Rearrange the fraction $\frac{1^{2} ((4 + x) (4 - x))}{4^{2} \cdot 1}$ .

$64 (- \sqrt{{(\frac{1}{4})}^{2} ((4 + x) (4 - x))} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{4}))) + C$

Step 12.1.11

Pull terms out from under the radical.

$64 (- (\frac{1}{4} \sqrt{(4 + x) (4 - x)}) + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{4}))) + C$

Step 12.1.12

Combine $\frac{1}{4}$ and $\sqrt{(4 + x) (4 - x)}$ .

$64 (- \frac{\sqrt{(4 + x) (4 - x)}}{4} + \frac{1}{3} \cos^{3} (\arcsin (\frac{x}{4}))) + C$

Step 12.2

Combine $\frac{1}{3}$ and $\cos^{3} (\arcsin (\frac{x}{4}))$ .

$64 (- \frac{\sqrt{(4 + x) (4 - x)}}{4} + \frac{\cos^{3} (\arcsin (\frac{x}{4}))}{3}) + C$

Step 12.3

Apply the distributive property.

$64 (- \frac{\sqrt{(4 + x) (4 - x)}}{4}) + 64 \frac{\cos^{3} (\arcsin (\frac{x}{4}))}{3} + C$

Step 12.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 12.4.1

Move the leading negative in $- \frac{\sqrt{(4 + x) (4 - x)}}{4}$ into the numerator.

$64 \frac{- \sqrt{(4 + x) (4 - x)}}{4} + 64 \frac{\cos^{3} (\arcsin (\frac{x}{4}))}{3} + C$

Step 12.4.2

Factor $4$ out of $64$ .

$4 (16) \frac{- \sqrt{(4 + x) (4 - x)}}{4} + 64 \frac{\cos^{3} (\arcsin (\frac{x}{4}))}{3} + C$

Step 12.4.3

Cancel the common factor.

$4 \cdot 16 \frac{- \sqrt{(4 + x) (4 - x)}}{4} + 64 \frac{\cos^{3} (\arcsin (\frac{x}{4}))}{3} + C$

Step 12.4.4

Rewrite the expression.

$16 (- \sqrt{(4 + x) (4 - x)}) + 64 \frac{\cos^{3} (\arcsin (\frac{x}{4}))}{3} + C$

Step 12.5

Multiply $- 1$ by $16$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\cos^{3} (\arcsin (\frac{x}{4}))}{3} + C$

Step 12.6

Simplify the numerator.

Tap for more steps...

Step 12.6.1

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{{\sqrt{1 - {(\frac{x}{4})}^{2}}}^{3}}{3} + C$

Step 12.6.2

Rewrite $1$ as $1^{2}$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{{\sqrt{1^{2} - {(\frac{x}{4})}^{2}}}^{3}}{3} + C$

Step 12.6.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \frac{x}{4}$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{{\sqrt{(1 + \frac{x}{4}) (1 - \frac{x}{4})}}^{3}}{3} + C$

Step 12.6.4

Write $1$ as a fraction with a common denominator.

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{{\sqrt{(\frac{4}{4} + \frac{x}{4}) (1 - \frac{x}{4})}}^{3}}{3} + C$

Step 12.6.5

Combine the numerators over the common denominator.

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{{\sqrt{\frac{4 + x}{4} (1 - \frac{x}{4})}}^{3}}{3} + C$

Step 12.6.6

Write $1$ as a fraction with a common denominator.

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{{\sqrt{\frac{4 + x}{4} (\frac{4}{4} - \frac{x}{4})}}^{3}}{3} + C$

Step 12.6.7

Combine the numerators over the common denominator.

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{{\sqrt{\frac{4 + x}{4} \cdot \frac{4 - x}{4}}}^{3}}{3} + C$

Step 12.6.8

Multiply $\frac{4 + x}{4}$ by $\frac{4 - x}{4}$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{{\sqrt{\frac{(4 + x) (4 - x)}{4 \cdot 4}}}^{3}}{3} + C$

Step 12.6.9

Multiply $4$ by $4$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{{\sqrt{\frac{(4 + x) (4 - x)}{16}}}^{3}}{3} + C$

Step 12.6.10

Rewrite $\frac{(4 + x) (4 - x)}{16}$ as ${(\frac{1}{4})}^{2} ((4 + x) (4 - x))$ .

Tap for more steps...

Step 12.6.10.1

Factor the perfect power $1^{2}$ out of $(4 + x) (4 - x)$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{{\sqrt{\frac{1^{2} ((4 + x) (4 - x))}{16}}}^{3}}{3} + C$

Step 12.6.10.2

Factor the perfect power $4^{2}$ out of $16$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{{\sqrt{\frac{1^{2} ((4 + x) (4 - x))}{4^{2} \cdot 1}}}^{3}}{3} + C$

Step 12.6.10.3

Rearrange the fraction $\frac{1^{2} ((4 + x) (4 - x))}{4^{2} \cdot 1}$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{{\sqrt{{(\frac{1}{4})}^{2} ((4 + x) (4 - x))}}^{3}}{3} + C$

Step 12.6.11

Pull terms out from under the radical.

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{{(\frac{1}{4} \sqrt{(4 + x) (4 - x)})}^{3}}{3} + C$

Step 12.6.12

Combine $\frac{1}{4}$ and $\sqrt{(4 + x) (4 - x)}$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{{(\frac{\sqrt{(4 + x) (4 - x)}}{4})}^{3}}{3} + C$

Step 12.6.13

Apply the product rule to $\frac{\sqrt{(4 + x) (4 - x)}}{4}$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{{\sqrt{(4 + x) (4 - x)}}^{3}}{4^{3}}}{3} + C$

Step 12.6.14

Simplify the numerator.

Tap for more steps...

Step 12.6.14.1

Rewrite ${\sqrt{(4 + x) (4 - x)}}^{3}$ as $\sqrt{{((4 + x) (4 - x))}^{3}}$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{\sqrt{{((4 + x) (4 - x))}^{3}}}{4^{3}}}{3} + C$

Step 12.6.14.2

Apply the product rule to $(4 + x) (4 - x)$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{\sqrt{{(4 + x)}^{3} {(4 - x)}^{3}}}{4^{3}}}{3} + C$

Step 12.6.14.3

Rewrite ${(4 + x)}^{3} {(4 - x)}^{3}$ as ${((4 + x) (4 - x))}^{2} ((4 + x) (4 - x))$ .

Tap for more steps...

Step 12.6.14.3.1

Factor out ${(4 + x)}^{2}$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{\sqrt{{(4 + x)}^{2} (4 + x) {(4 - x)}^{3}}}{4^{3}}}{3} + C$

Step 12.6.14.3.2

Factor out ${(4 - x)}^{2}$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{\sqrt{{(4 + x)}^{2} (4 + x) ({(4 - x)}^{2} (4 - x))}}{4^{3}}}{3} + C$

Step 12.6.14.3.3

Move $4 + x$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{\sqrt{{(4 + x)}^{2} {(4 - x)}^{2} (4 + x) (4 - x)}}{4^{3}}}{3} + C$

Step 12.6.14.3.4

Rewrite ${(4 + x)}^{2} {(4 - x)}^{2}$ as ${((4 + x) (4 - x))}^{2}$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{\sqrt{{((4 + x) (4 - x))}^{2} (4 + x) (4 - x)}}{4^{3}}}{3} + C$

Step 12.6.14.3.5

Add parentheses.

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{\sqrt{{((4 + x) (4 - x))}^{2} ((4 + x) (4 - x))}}{4^{3}}}{3} + C$

Step 12.6.14.4

Pull terms out from under the radical.

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{(4 + x) (4 - x) \sqrt{(4 + x) (4 - x)}}{4^{3}}}{3} + C$

Step 12.6.14.5

Expand $(4 + x) (4 - x)$ using the FOIL Method.

Tap for more steps...

Step 12.6.14.5.1

Apply the distributive property.

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{(4 (4 - x) + x (4 - x)) \sqrt{(4 + x) (4 - x)}}{4^{3}}}{3} + C$

Step 12.6.14.5.2

Apply the distributive property.

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{(4 \cdot 4 + 4 (- x) + x (4 - x)) \sqrt{(4 + x) (4 - x)}}{4^{3}}}{3} + C$

Step 12.6.14.5.3

Apply the distributive property.

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{(4 \cdot 4 + 4 (- x) + x \cdot 4 + x (- x)) \sqrt{(4 + x) (4 - x)}}{4^{3}}}{3} + C$

Step 12.6.14.6

Simplify and combine like terms.

Tap for more steps...

Step 12.6.14.6.1

Simplify each term.

Tap for more steps...

Step 12.6.14.6.1.1

Multiply $4$ by $4$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{(16 + 4 (- x) + x \cdot 4 + x (- x)) \sqrt{(4 + x) (4 - x)}}{4^{3}}}{3} + C$

Step 12.6.14.6.1.2

Multiply $- 1$ by $4$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{(16 - 4 x + x \cdot 4 + x (- x)) \sqrt{(4 + x) (4 - x)}}{4^{3}}}{3} + C$

Step 12.6.14.6.1.3

Move $4$ to the left of $x$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{(16 - 4 x + 4 \cdot x + x (- x)) \sqrt{(4 + x) (4 - x)}}{4^{3}}}{3} + C$

Step 12.6.14.6.1.4

Rewrite using the commutative property of multiplication.

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{(16 - 4 x + 4 x - x \cdot x) \sqrt{(4 + x) (4 - x)}}{4^{3}}}{3} + C$

Step 12.6.14.6.1.5

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 12.6.14.6.1.5.1

Move $x$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{(16 - 4 x + 4 x - (x \cdot x)) \sqrt{(4 + x) (4 - x)}}{4^{3}}}{3} + C$

Step 12.6.14.6.1.5.2

Multiply $x$ by $x$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{(16 - 4 x + 4 x - x^{2}) \sqrt{(4 + x) (4 - x)}}{4^{3}}}{3} + C$

Step 12.6.14.6.2

Add $- 4 x$ and $4 x$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{(16 + 0 - x^{2}) \sqrt{(4 + x) (4 - x)}}{4^{3}}}{3} + C$

Step 12.6.14.6.3

Add $16$ and $0$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{(16 - x^{2}) \sqrt{(4 + x) (4 - x)}}{4^{3}}}{3} + C$

Step 12.6.14.7

Apply the distributive property.

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{16 \sqrt{(4 + x) (4 - x)} - x^{2} \sqrt{(4 + x) (4 - x)}}{4^{3}}}{3} + C$

Step 12.6.14.8

Factor $\sqrt{(4 + x) (4 - x)}$ out of $16 \sqrt{(4 + x) (4 - x)} - x^{2} \sqrt{(4 + x) (4 - x)}$ .

Tap for more steps...

Step 12.6.14.8.1

Factor $\sqrt{(4 + x) (4 - x)}$ out of $16 \sqrt{(4 + x) (4 - x)}$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{\sqrt{(4 + x) (4 - x)} \cdot 16 - x^{2} \sqrt{(4 + x) (4 - x)}}{4^{3}}}{3} + C$

Step 12.6.14.8.2

Factor $\sqrt{(4 + x) (4 - x)}$ out of $- x^{2} \sqrt{(4 + x) (4 - x)}$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{\sqrt{(4 + x) (4 - x)} \cdot 16 + \sqrt{(4 + x) (4 - x)} (- x^{2})}{4^{3}}}{3} + C$

Step 12.6.14.8.3

Factor $\sqrt{(4 + x) (4 - x)}$ out of $\sqrt{(4 + x) (4 - x)} \cdot 16 + \sqrt{(4 + x) (4 - x)} (- x^{2})$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{\sqrt{(4 + x) (4 - x)} (16 - x^{2})}{4^{3}}}{3} + C$

Step 12.6.14.9

Rewrite $16$ as $4^{2}$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{\sqrt{(4 + x) (4 - x)} (4^{2} - x^{2})}{4^{3}}}{3} + C$

Step 12.6.14.10

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = x$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{\sqrt{(4 + x) (4 - x)} (4 + x) (4 - x)}{4^{3}}}{3} + C$

Step 12.6.15

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

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\frac{\sqrt{(4 + x) (4 - x)} (4 + x) (4 - x)}{64}}{3} + C$

Step 12.7

Simplify each term.

Tap for more steps...

Step 12.7.1

Multiply the numerator by the reciprocal of the denominator.

$- 16 \sqrt{(4 + x) (4 - x)} + 64 (\frac{\sqrt{(4 + x) (4 - x)} (4 + x) (4 - x)}{64} \cdot \frac{1}{3}) + C$

Step 12.7.2

Multiply $\frac{\sqrt{(4 + x) (4 - x)} (4 + x) (4 - x)}{64} \cdot \frac{1}{3}$ .

Tap for more steps...

Step 12.7.2.1

Multiply $\frac{\sqrt{(4 + x) (4 - x)} (4 + x) (4 - x)}{64}$ by $\frac{1}{3}$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\sqrt{(4 + x) (4 - x)} (4 + x) (4 - x)}{64 \cdot 3} + C$

Step 12.7.2.2

Multiply $64$ by $3$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\sqrt{(4 + x) (4 - x)} (4 + x) (4 - x)}{192} + C$

Step 12.7.3

Cancel the common factor of $64$ .

Tap for more steps...

Step 12.7.3.1

Factor $64$ out of $192$ .

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\sqrt{(4 + x) (4 - x)} (4 + x) (4 - x)}{64 (3)} + C$

Step 12.7.3.2

Cancel the common factor.

$- 16 \sqrt{(4 + x) (4 - x)} + 64 \frac{\sqrt{(4 + x) (4 - x)} (4 + x) (4 - x)}{64 \cdot 3} + C$

Step 12.7.3.3

Rewrite the expression.

$- 16 \sqrt{(4 + x) (4 - x)} + \frac{\sqrt{(4 + x) (4 - x)} (4 + x) (4 - x)}{3} + C$

Step 12.8

To write $- 16 \sqrt{(4 + x) (4 - x)}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 16 \sqrt{(4 + x) (4 - x)} \cdot \frac{3}{3} + \frac{\sqrt{(4 + x) (4 - x)} (4 + x) (4 - x)}{3} + C$

Step 12.9

Combine $- 16 \sqrt{(4 + x) (4 - x)}$ and $\frac{3}{3}$ .

$\frac{- 16 \sqrt{(4 + x) (4 - x)} \cdot 3}{3} + \frac{\sqrt{(4 + x) (4 - x)} (4 + x) (4 - x)}{3} + C$

Step 12.10

Combine the numerators over the common denominator.

$\frac{- 16 \sqrt{(4 + x) (4 - x)} \cdot 3 + \sqrt{(4 + x) (4 - x)} (4 + x) (4 - x)}{3} + C$

Step 12.11

Simplify the numerator.

Tap for more steps...

Step 12.11.1

Factor $\sqrt{(4 + x) (4 - x)}$ out of $- 16 \sqrt{(4 + x) (4 - x)} \cdot 3 + \sqrt{(4 + x) (4 - x)} (4 + x) (4 - x)$ .

Tap for more steps...

Step 12.11.1.1

Factor $\sqrt{(4 + x) (4 - x)}$ out of $- 16 \sqrt{(4 + x) (4 - x)} \cdot 3$ .

$\frac{\sqrt{(4 + x) (4 - x)} (- 16 \cdot 3) + \sqrt{(4 + x) (4 - x)} (4 + x) (4 - x)}{3} + C$

Step 12.11.1.2

Factor $\sqrt{(4 + x) (4 - x)}$ out of $\sqrt{(4 + x) (4 - x)} (4 + x) (4 - x)$ .

$\frac{\sqrt{(4 + x) (4 - x)} (- 16 \cdot 3) + \sqrt{(4 + x) (4 - x)} ((4 + x) (4 - x))}{3} + C$

Step 12.11.1.3

Factor $\sqrt{(4 + x) (4 - x)}$ out of $\sqrt{(4 + x) (4 - x)} (- 16 \cdot 3) + \sqrt{(4 + x) (4 - x)} ((4 + x) (4 - x))$ .

$\frac{\sqrt{(4 + x) (4 - x)} (- 16 \cdot 3 + (4 + x) (4 - x))}{3} + C$

Step 12.11.2

Multiply $- 16$ by $3$ .

$\frac{\sqrt{(4 + x) (4 - x)} (- 48 + (4 + x) (4 - x))}{3} + C$

Step 12.11.3

Expand $(4 + x) (4 - x)$ using the FOIL Method.

Tap for more steps...

Step 12.11.3.1

Apply the distributive property.

$\frac{\sqrt{(4 + x) (4 - x)} (- 48 + 4 (4 - x) + x (4 - x))}{3} + C$

Step 12.11.3.2

Apply the distributive property.

$\frac{\sqrt{(4 + x) (4 - x)} (- 48 + 4 \cdot 4 + 4 (- x) + x (4 - x))}{3} + C$

Step 12.11.3.3

Apply the distributive property.

$\frac{\sqrt{(4 + x) (4 - x)} (- 48 + 4 \cdot 4 + 4 (- x) + x \cdot 4 + x (- x))}{3} + C$

Step 12.11.4

Simplify and combine like terms.

Tap for more steps...

Step 12.11.4.1

Simplify each term.

Tap for more steps...

Step 12.11.4.1.1

Multiply $4$ by $4$ .

$\frac{\sqrt{(4 + x) (4 - x)} (- 48 + 16 + 4 (- x) + x \cdot 4 + x (- x))}{3} + C$

Step 12.11.4.1.2

Multiply $- 1$ by $4$ .

$\frac{\sqrt{(4 + x) (4 - x)} (- 48 + 16 - 4 x + x \cdot 4 + x (- x))}{3} + C$

Step 12.11.4.1.3

Move $4$ to the left of $x$ .

$\frac{\sqrt{(4 + x) (4 - x)} (- 48 + 16 - 4 x + 4 \cdot x + x (- x))}{3} + C$

Step 12.11.4.1.4

Rewrite using the commutative property of multiplication.

$\frac{\sqrt{(4 + x) (4 - x)} (- 48 + 16 - 4 x + 4 x - x \cdot x)}{3} + C$

Step 12.11.4.1.5

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 12.11.4.1.5.1

Move $x$ .

$\frac{\sqrt{(4 + x) (4 - x)} (- 48 + 16 - 4 x + 4 x - (x \cdot x))}{3} + C$

Step 12.11.4.1.5.2

Multiply $x$ by $x$ .

$\frac{\sqrt{(4 + x) (4 - x)} (- 48 + 16 - 4 x + 4 x - x^{2})}{3} + C$

Step 12.11.4.2

Add $- 4 x$ and $4 x$ .

$\frac{\sqrt{(4 + x) (4 - x)} (- 48 + 16 + 0 - x^{2})}{3} + C$

Step 12.11.4.3

Add $16$ and $0$ .

$\frac{\sqrt{(4 + x) (4 - x)} (- 48 + 16 - x^{2})}{3} + C$

Step 12.11.5

Add $- 48$ and $16$ .

$\frac{\sqrt{(4 + x) (4 - x)} (- 32 - x^{2})}{3} + C$

Step 12.12

Rewrite $- 32$ as $- 1 (32)$ .

$\frac{\sqrt{(4 + x) (4 - x)} (- 1 (32) - x^{2})}{3} + C$

Step 12.13

Factor $- 1$ out of $- x^{2}$ .

$\frac{\sqrt{(4 + x) (4 - x)} (- 1 (32) - (x^{2}))}{3} + C$

Step 12.14

Factor $- 1$ out of $- 1 (32) - (x^{2})$ .

$\frac{\sqrt{(4 + x) (4 - x)} (- 1 (32 + x^{2}))}{3} + C$

Step 12.15

Move the negative in front of the fraction.

$- \frac{\sqrt{(4 + x) (4 - x)} (32 + x^{2})}{3} + C$