Calculus Examples

$\int_{0}^{8} \frac{1}{\sqrt[4]{1 + x}} d x$

Step 1

Let $u = 1 + x$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 1 + x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $1 + x$ .

$\frac{d}{d x} [1 + x]$

Step 1.1.2

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

$\frac{d}{d x} [1] + \frac{d}{d x} [x]$

Step 1.1.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x]$

Step 1.1.4

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

$0 + 1$

Step 1.1.5

Add $0$ and $1$ .

$1$

Step 1.2

Substitute the lower limit in for $x$ in $u = 1 + x$ .

$u_{lower} = 1 + 0$

Step 1.3

Add $1$ and $0$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = 1 + x$ .

$u_{upper} = 1 + 8$

Step 1.5

Add $1$ and $8$ .

$u_{upper} = 9$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 9$

Step 1.7

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

$\int_{1}^{9} \frac{1}{\sqrt[4]{u}} d u$

Step 2

Apply basic rules of exponents.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{u}$ as $u^{\frac{1}{4}}$ .

$\int_{1}^{9} \frac{1}{u^{\frac{1}{4}}} d u$

Step 2.2

Move $u^{\frac{1}{4}}$ out of the denominator by raising it to the $- 1$ power.

$\int_{1}^{9} {(u^{\frac{1}{4}})}^{- 1} d u$

Step 2.3

Multiply the exponents in ${(u^{\frac{1}{4}})}^{- 1}$ .

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Step 2.3.1

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

$\int_{1}^{9} u^{\frac{1}{4} \cdot - 1} d u$

Step 2.3.2

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

$\int_{1}^{9} u^{\frac{- 1}{4}} d u$

Step 2.3.3

Move the negative in front of the fraction.

$\int_{1}^{9} u^{- \frac{1}{4}} d u$

Step 3

By the Power Rule, the integral of $u^{- \frac{1}{4}}$ with respect to $u$ is $\frac{4}{3} u^{\frac{3}{4}}$ .

$\frac{4}{3} u^{\frac{3}{4}}]_{1}^{9}$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $\frac{4}{3} u^{\frac{3}{4}}$ at $9$ and at $1$ .

$(\frac{4}{3} \cdot 9^{\frac{3}{4}}) - \frac{4}{3} \cdot 1^{\frac{3}{4}}$

Step 4.2

Simplify.

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Step 4.2.1

Combine $\frac{4}{3}$ and $9^{\frac{3}{4}}$ .

$\frac{4 \cdot 9^{\frac{3}{4}}}{3} - \frac{4}{3} \cdot 1^{\frac{3}{4}}$

Step 4.2.2

One to any power is one.

$\frac{4 \cdot 9^{\frac{3}{4}}}{3} - \frac{4}{3} \cdot 1$

Step 4.2.3

Multiply $- 1$ by $1$ .

$\frac{4 \cdot 9^{\frac{3}{4}}}{3} - \frac{4}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{4 \cdot 9^{\frac{3}{4}}}{3} - \frac{4}{3}$

Decimal Form:

$5.59486989 \dots$

Step 6