Calculus Examples

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

Step 1

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

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

Step 2

Let $u = 1 - 4 x^{2}$ . 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 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Differentiate.

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Step 2.1.2.1

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 2.1.2.2

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 2.1.3

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

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Step 2.1.3.1

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 2.1.3.2

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 2.1.3.3

Multiply $2$ by $- 4$ .

$0 - 8 x$

Step 2.1.4

Subtract $8 x$ from $0$ .

$- 8 x$

Step 2.2

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

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

Step 2.3

Simplify.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

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

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

Step 2.3.1.2

Multiply $- 4$ by $0$ .

$u_{lower} = 1 + 0$

Step 2.3.2

Add $1$ and $0$ .

$u_{lower} = 1$

Step 2.4

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

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

Step 2.5

Simplify.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

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

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

Step 2.5.1.2

One to any power is one.

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

Step 2.5.1.3

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

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

Step 2.5.1.4

Cancel the common factor of $4$ .

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Step 2.5.1.4.1

Factor $4$ out of $- 4$ .

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

Step 2.5.1.4.2

Factor $4$ out of $16$ .

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

Step 2.5.1.4.3

Cancel the common factor.

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

Step 2.5.1.4.4

Rewrite the expression.

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

Step 2.5.1.5

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

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

Step 2.5.2

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

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

Step 2.5.3

Combine the numerators over the common denominator.

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

Step 2.5.4

Subtract $1$ from $4$ .

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

Step 2.6

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 2.7

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

$8 \int_{1}^{\frac{3}{4}} \frac{1}{\sqrt{u}} \cdot \frac{1}{- 8} d u$

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

$8 \int_{1}^{\frac{3}{4}} \frac{1}{\sqrt{u}} (- \frac{1}{8}) d u$

Step 3.2

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

$8 \int_{1}^{\frac{3}{4}} - \frac{1}{\sqrt{u} \cdot 8} d u$

Step 3.3

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

$8 \int_{1}^{\frac{3}{4}} - \frac{1}{8 \sqrt{u}} d u$

Step 4

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

$8 (- \int_{1}^{\frac{3}{4}} \frac{1}{8 \sqrt{u}} d u)$

Step 5

Multiply $- 1$ by $8$ .

$- 8 \int_{1}^{\frac{3}{4}} \frac{1}{8 \sqrt{u}} d u$

Step 6

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

$- 8 (\frac{1}{8} \int_{1}^{\frac{3}{4}} \frac{1}{\sqrt{u}} d u)$

Step 7

Simplify the expression.

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Step 7.1

Simplify.

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Step 7.1.1

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

$\frac{- 8}{8} \int_{1}^{\frac{3}{4}} \frac{1}{\sqrt{u}} d u$

Step 7.1.2

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

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Step 7.1.2.1

Factor $8$ out of $- 8$ .

$\frac{8 \cdot - 1}{8} \int_{1}^{\frac{3}{4}} \frac{1}{\sqrt{u}} d u$

Step 7.1.2.2

Cancel the common factors.

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Step 7.1.2.2.1

Factor $8$ out of $8$ .

$\frac{8 \cdot - 1}{8 (1)} \int_{1}^{\frac{3}{4}} \frac{1}{\sqrt{u}} d u$

Step 7.1.2.2.2

Cancel the common factor.

$\frac{8 \cdot - 1}{8 \cdot 1} \int_{1}^{\frac{3}{4}} \frac{1}{\sqrt{u}} d u$

Step 7.1.2.2.3

Rewrite the expression.

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

Step 7.1.2.2.4

Divide $- 1$ by $1$ .

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

Step 7.2

Apply basic rules of exponents.

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Step 7.2.1

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 7.2.2

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 7.2.3

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

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Step 7.2.3.1

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 7.2.3.2

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

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

Step 7.2.3.3

Move the negative in front of the fraction.

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

Step 8

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 9

Substitute and simplify.

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Step 9.1

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 9.2

Simplify.

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Step 9.2.1

One to any power is one.

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

Step 9.2.2

Multiply $- 2$ by $1$ .

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

Step 10

Simplify.

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Step 10.1

Simplify each term.

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Step 10.1.1

Apply the product rule to $\frac{3}{4}$ .

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

Step 10.1.2

Simplify the denominator.

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Step 10.1.2.1

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

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

Step 10.1.2.2

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

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

Step 10.1.2.3

Cancel the common factor of $2$ .

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Step 10.1.2.3.1

Cancel the common factor.

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

Step 10.1.2.3.2

Rewrite the expression.

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

Step 10.1.2.4

Evaluate the exponent.

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

Step 10.1.3

Cancel the common factor of $2$ .

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Step 10.1.3.1

Cancel the common factor.

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

Step 10.1.3.2

Rewrite the expression.

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

Step 10.2

Apply the distributive property.

$- 3^{\frac{1}{2}} - - 2$

Step 10.3

Multiply $- 1$ by $- 2$ .

$- 3^{\frac{1}{2}} + 2$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- 3^{\frac{1}{2}} + 2$

Decimal Form:

$0.26794919 \dots$

Step 12