Calculus Examples

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

Step 1

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

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Step 1.1

Let $u = 1 - 4 x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $1 - 4 x^{2}$ .

$\frac{d}{d x} [1 - 4 x^{2}]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $1 - 4 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 4 x^{2}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [- 4 x^{2}]$

Step 1.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ 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 $u = 1 - 4 x^{2}$ .

$u_{lower} = 1 - 4 \cdot 0^{2}$

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 $0$ .

$u_{lower} = 1 - 4 \cdot 0$

Step 1.3.1.2

Multiply $- 4$ by $0$ .

$u_{lower} = 1 + 0$

Step 1.3.2

Add $1$ and $0$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = 1 - 4 x^{2}$ .

$u_{upper} = 1 - 4 {(\frac{1}{4})}^{2}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Apply the product rule to $\frac{1}{4}$ .

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

Step 1.5.1.2

One to any power is one.

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

Step 1.5.1.3

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

$u_{upper} = 1 - 4 (\frac{1}{16})$

Step 1.5.1.4

Cancel the common factor of $4$ .

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Step 1.5.1.4.1

Factor $4$ out of $- 4$ .

$u_{upper} = 1 + 4 (- 1) \frac{1}{16}$

Step 1.5.1.4.2

Factor $4$ out of $16$ .

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

Step 1.5.1.4.3

Cancel the common factor.

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

Step 1.5.1.4.4

Rewrite the expression.

$u_{upper} = 1 - 1 (\frac{1}{4})$

Step 1.5.1.5

Rewrite $- 1 (\frac{1}{4})$ as $- (\frac{1}{4})$ .

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

Step 1.5.2

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

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

Step 1.5.3

Combine the numerators over the common denominator.

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

Step 1.5.4

Subtract $1$ from $4$ .

$u_{upper} = \frac{3}{4}$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = \frac{3}{4}$

Step 1.7

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

$\int_{1}^{\frac{3}{4}} \frac{- 1}{\sqrt{u}} d u$

Step 2

Move the negative in front of the fraction.

$\int_{1}^{\frac{3}{4}} - \frac{1}{\sqrt{u}} d u$

Step 3

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

$- \int_{1}^{\frac{3}{4}} \frac{1}{\sqrt{u}} d u$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

$- \int_{1}^{\frac{3}{4}} \frac{1}{u^{\frac{1}{2}}} d u$

Step 4.2

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

$- \int_{1}^{\frac{3}{4}} {(u^{\frac{1}{2}})}^{- 1} d u$

Step 4.3

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

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Step 4.3.1

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

$- \int_{1}^{\frac{3}{4}} u^{\frac{1}{2} \cdot - 1} d u$

Step 4.3.2

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

$- \int_{1}^{\frac{3}{4}} u^{\frac{- 1}{2}} d u$

Step 4.3.3

Move the negative in front of the fraction.

$- \int_{1}^{\frac{3}{4}} u^{- \frac{1}{2}} d u$

Step 5

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

$- (2 u^{\frac{1}{2}}]_{1}^{\frac{3}{4}})$

Step 6

Simplify the expression.

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Step 6.1

Evaluate $2 u^{\frac{1}{2}}$ at $\frac{3}{4}$ and at $1$ .

$- ((2 {(\frac{3}{4})}^{\frac{1}{2}}) - 2 \cdot 1^{\frac{1}{2}})$

Step 6.2

One to any power is one.

$- (2 {(\frac{3}{4})}^{\frac{1}{2}} - 2 \cdot 1)$

Step 6.3

Multiply $- 2$ by $1$ .

$- (2 {(\frac{3}{4})}^{\frac{1}{2}} - 2)$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- (2 {(\frac{3}{4})}^{\frac{1}{2}} - 2)$

Decimal Form:

$0.26794919 \dots$