Calculus Examples

$\int_{0}^{1} \frac{x}{\sqrt{4 - 4 x^{2}}} d x$

Step 1

Let $u = 4 - 4 x^{2}$ . Then $d u = - 8 x d x$ , so $- \frac{1}{8} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 4 - 4 x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $4 - 4 x^{2}$ .

$\frac{d}{d x} [4 - 4 x^{2}]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $4 - 4 x^{2}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- 4 x^{2}]$ .

$\frac{d}{d x} [4] + \frac{d}{d x} [- 4 x^{2}]$

Step 1.1.2.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 4 x^{2}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 4 x^{2}]$ .

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Step 1.1.3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{2}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{2}]$ .

$0 - 4 \frac{d}{d x} [x^{2}]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 - 4 (2 x)$

Step 1.1.3.3

Multiply $2$ by $- 4$ .

$0 - 8 x$

Step 1.1.4

Subtract $8 x$ from $0$ .

$- 8 x$

Step 1.2

Substitute the lower limit in for $x$ in $u = 4 - 4 x^{2}$ .

$u_{lower} = 4 - 4 \cdot 0^{2}$

Step 1.3

Simplify.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Raising $0$ to any positive power yields $0$ .

$u_{lower} = 4 - 4 \cdot 0$

Step 1.3.1.2

Multiply $- 4$ by $0$ .

$u_{lower} = 4 + 0$

Step 1.3.2

Add $4$ and $0$ .

$u_{lower} = 4$

Step 1.4

Substitute the upper limit in for $x$ in $u = 4 - 4 x^{2}$ .

$u_{upper} = 4 - 4 \cdot 1^{2}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

One to any power is one.

$u_{upper} = 4 - 4 \cdot 1$

Step 1.5.1.2

Multiply $- 4$ by $1$ .

$u_{upper} = 4 - 4$

Step 1.5.2

Subtract $4$ from $4$ .

$u_{upper} = 0$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 4$

$u_{upper} = 0$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{4}^{0} \frac{1}{\sqrt{u}} \cdot \frac{1}{- 8} d u$

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

$\int_{4}^{0} \frac{1}{\sqrt{u}} (- \frac{1}{8}) d u$

Step 2.2

Multiply $\frac{1}{\sqrt{u}}$ by $\frac{1}{8}$ .

$\int_{4}^{0} - \frac{1}{\sqrt{u} \cdot 8} d u$

Step 2.3

Move $8$ to the left of $\sqrt{u}$ .

$\int_{4}^{0} - \frac{1}{8 \sqrt{u}} d u$

Step 3

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \int_{4}^{0} \frac{1}{8 \sqrt{u}} d u$

Step 4

Since $\frac{1}{8}$ is constant with respect to $u$ , move $\frac{1}{8}$ out of the integral.

$- (\frac{1}{8} \int_{4}^{0} \frac{1}{\sqrt{u}} d u)$

Step 5

Apply basic rules of exponents.

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Step 5.1

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

$- \frac{1}{8} \int_{4}^{0} \frac{1}{u^{\frac{1}{2}}} d u$

Step 5.2

Move $u^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$- \frac{1}{8} \int_{4}^{0} {(u^{\frac{1}{2}})}^{- 1} d u$

Step 5.3

Multiply the exponents in ${(u^{\frac{1}{2}})}^{- 1}$ .

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Step 5.3.1

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

$- \frac{1}{8} \int_{4}^{0} u^{\frac{1}{2} \cdot - 1} d u$

Step 5.3.2

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

$- \frac{1}{8} \int_{4}^{0} u^{\frac{- 1}{2}} d u$

Step 5.3.3

Move the negative in front of the fraction.

$- \frac{1}{8} \int_{4}^{0} u^{- \frac{1}{2}} d u$

Step 6

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

$- \frac{1}{8} 2 u^{\frac{1}{2}}]_{4}^{0}$

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $2 u^{\frac{1}{2}}$ at $0$ and at $4$ .

$- \frac{1}{8} ((2 \cdot 0^{\frac{1}{2}}) - 2 \cdot 4^{\frac{1}{2}})$

Step 7.2

Simplify.

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Step 7.2.1

Rewrite $0$ as $0^{2}$ .

$- \frac{1}{8} (2 \cdot {(0^{2})}^{\frac{1}{2}} - 2 \cdot 4^{\frac{1}{2}})$

Step 7.2.2

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

$- \frac{1}{8} (2 \cdot 0^{2 (\frac{1}{2})} - 2 \cdot 4^{\frac{1}{2}})$

Step 7.2.3

Cancel the common factor of $2$ .

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Step 7.2.3.1

Cancel the common factor.

$- \frac{1}{8} (2 \cdot 0^{2 (\frac{1}{2})} - 2 \cdot 4^{\frac{1}{2}})$

Step 7.2.3.2

Rewrite the expression.

$- \frac{1}{8} (2 \cdot 0^{1} - 2 \cdot 4^{\frac{1}{2}})$

Step 7.2.4

Evaluate the exponent.

$- \frac{1}{8} (2 \cdot 0 - 2 \cdot 4^{\frac{1}{2}})$

Step 7.2.5

Multiply $2$ by $0$ .

$- \frac{1}{8} (0 - 2 \cdot 4^{\frac{1}{2}})$

Step 7.2.6

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

$- \frac{1}{8} (0 - 2 \cdot {(2^{2})}^{\frac{1}{2}})$

Step 7.2.7

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

$- \frac{1}{8} (0 - 2 \cdot 2^{2 (\frac{1}{2})})$

Step 7.2.8

Cancel the common factor of $2$ .

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Step 7.2.8.1

Cancel the common factor.

$- \frac{1}{8} (0 - 2 \cdot 2^{2 (\frac{1}{2})})$

Step 7.2.8.2

Rewrite the expression.

$- \frac{1}{8} (0 - 2 \cdot 2^{1})$

Step 7.2.9

Evaluate the exponent.

$- \frac{1}{8} (0 - 2 \cdot 2)$

Step 7.2.10

Multiply $- 2$ by $2$ .

$- \frac{1}{8} (0 - 4)$

Step 7.2.11

Subtract $4$ from $0$ .

$- \frac{1}{8} \cdot - 4$

Step 7.2.12

Multiply $- 4$ by $- 1$ .

$4 (\frac{1}{8})$

Step 7.2.13

Combine $4$ and $\frac{1}{8}$ .

$\frac{4}{8}$

Step 7.2.14

Cancel the common factor of $4$ and $8$ .

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Step 7.2.14.1

Factor $4$ out of $4$ .

$\frac{4 (1)}{8}$

Step 7.2.14.2

Cancel the common factors.

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Step 7.2.14.2.1

Factor $4$ out of $8$ .

$\frac{4 \cdot 1}{4 \cdot 2}$

Step 7.2.14.2.2

Cancel the common factor.

$\frac{4 \cdot 1}{4 \cdot 2}$

Step 7.2.14.2.3

Rewrite the expression.

$\frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$

Step 9