Calculus Examples

$\int_{0}^{1} x^{2} \cdot \sqrt{x^{3} + 2} d x$

Step 1

Let $u = x^{3} + 2$ . Then $d u = 3 x^{2} d x$ , so $\frac{1}{3} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x^{3} + 2$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{3} + 2$ .

$\frac{d}{d x} [x^{3} + 2]$

Step 1.1.2

By the Sum Rule, the derivative of $x^{3} + 2$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [2]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [2]$

Step 1.1.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$3 x^{2} + 0$

Step 1.1.5

Add $3 x^{2}$ and $0$ .

$3 x^{2}$

Step 1.2

Substitute the lower limit in for $x$ in $u = x^{3} + 2$ .

$u_{lower} = 0^{3} + 2$

Step 1.3

Simplify.

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Step 1.3.1

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

$u_{lower} = 0 + 2$

Step 1.3.2

Add $0$ and $2$ .

$u_{lower} = 2$

Step 1.4

Substitute the upper limit in for $x$ in $u = x^{3} + 2$ .

$u_{upper} = 1^{3} + 2$

Step 1.5

Simplify.

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Step 1.5.1

One to any power is one.

$u_{upper} = 1 + 2$

Step 1.5.2

Add $1$ and $2$ .

$u_{upper} = 3$

Step 1.6

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

$u_{lower} = 2$

$u_{upper} = 3$

Step 1.7

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

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

Step 2

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

$\int_{2}^{3} \frac{\sqrt{u}}{3} d u$

Step 3

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

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

Step 4

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

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

Step 5

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

$\frac{1}{3} \frac{2}{3} u^{\frac{3}{2}}]_{2}^{3}$

Step 6

Substitute and simplify.

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Step 6.1

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

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

Step 6.2

Simplify.

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Step 6.2.1

Combine $\frac{2}{3}$ and $3^{\frac{3}{2}}$ .

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

Step 6.2.2

Move $3^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 6.2.3

Multiply $3^{\frac{3}{2}}$ by $3^{- 1}$ by adding the exponents.

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Step 6.2.3.1

Move $3^{- 1}$ .

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

Step 6.2.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.2.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6.2.3.4

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

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

Step 6.2.3.5

Combine the numerators over the common denominator.

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

Step 6.2.3.6

Simplify the numerator.

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Step 6.2.3.6.1

Multiply $- 1$ by $2$ .

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

Step 6.2.3.6.2

Add $- 2$ and $3$ .

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

Step 6.2.4

Combine $2^{\frac{3}{2}}$ and $\frac{2}{3}$ .

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

Step 6.2.5

Multiply $2^{\frac{3}{2}}$ by $2$ by adding the exponents.

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Step 6.2.5.1

Multiply $2^{\frac{3}{2}}$ by $2$ .

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Step 6.2.5.1.1

Raise $2$ to the power of $1$ .

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

Step 6.2.5.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.2.5.2

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

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

Step 6.2.5.3

Combine the numerators over the common denominator.

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

Step 6.2.5.4

Add $3$ and $2$ .

$\frac{1}{3} (2 \cdot 3^{\frac{1}{2}} - \frac{2^{\frac{5}{2}}}{3})$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$\frac{1}{3} (2 \cdot 3^{\frac{1}{2}}) + \frac{1}{3} (- \frac{2^{\frac{5}{2}}}{3})$

Step 7.2

Multiply $\frac{1}{3} (2 \cdot 3^{\frac{1}{2}})$ .

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Step 7.2.1

Combine $2$ and $\frac{1}{3}$ .

$\frac{2}{3} \cdot 3^{\frac{1}{2}} + \frac{1}{3} (- \frac{2^{\frac{5}{2}}}{3})$

Step 7.2.2

Combine $\frac{2}{3}$ and $3^{\frac{1}{2}}$ .

$\frac{2 \cdot 3^{\frac{1}{2}}}{3} + \frac{1}{3} (- \frac{2^{\frac{5}{2}}}{3})$

Step 7.3

Multiply $\frac{1}{3} (- \frac{2^{\frac{5}{2}}}{3})$ .

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Step 7.3.1

Multiply $\frac{1}{3}$ by $\frac{2^{\frac{5}{2}}}{3}$ .

$\frac{2 \cdot 3^{\frac{1}{2}}}{3} - \frac{2^{\frac{5}{2}}}{3 \cdot 3}$

Step 7.3.2

Multiply $3$ by $3$ .

$\frac{2 \cdot 3^{\frac{1}{2}}}{3} - \frac{2^{\frac{5}{2}}}{9}$

Step 7.4

Simplify each term.

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Step 7.4.1

Move $3^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{2}{3 \cdot 3^{- \frac{1}{2}}} - \frac{2^{\frac{5}{2}}}{9}$

Step 7.4.2

Multiply $3$ by $3^{- \frac{1}{2}}$ by adding the exponents.

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Step 7.4.2.1

Multiply $3$ by $3^{- \frac{1}{2}}$ .

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Step 7.4.2.1.1

Raise $3$ to the power of $1$ .

$\frac{2}{3^{1} \cdot 3^{- \frac{1}{2}}} - \frac{2^{\frac{5}{2}}}{9}$

Step 7.4.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2}{3^{1 - \frac{1}{2}}} - \frac{2^{\frac{5}{2}}}{9}$

Step 7.4.2.2

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

$\frac{2}{3^{\frac{2}{2} - \frac{1}{2}}} - \frac{2^{\frac{5}{2}}}{9}$

Step 7.4.2.3

Combine the numerators over the common denominator.

$\frac{2}{3^{\frac{2 - 1}{2}}} - \frac{2^{\frac{5}{2}}}{9}$

Step 7.4.2.4

Subtract $1$ from $2$ .

$\frac{2}{3^{\frac{1}{2}}} - \frac{2^{\frac{5}{2}}}{9}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3^{\frac{1}{2}}} - \frac{2^{\frac{5}{2}}}{9}$

Decimal Form:

$0.52616117 \dots$

Step 9