Calculus Examples

$\int_{0}^{1} (x + 2) {(3 x^{2} + 12 x + 1)}^{\frac{1}{2}} d x$

Step 1

Let $u = 3 x^{2} + 12 x + 1$ . Then $d u = (6 x + 12) d x$ , so $\frac{1}{6} d u = (x + 2) d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 3 x^{2} + 12 x + 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $3 x^{2} + 12 x + 1$ .

$\frac{d}{d x} [3 x^{2} + 12 x + 1]$

Step 1.1.2

By the Sum Rule, the derivative of $3 x^{2} + 12 x + 1$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [1]$

Step 1.1.3

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

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Step 1.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [1]$

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 $n = 2$ .

$3 (2 x) + \frac{d}{d x} [12 x] + \frac{d}{d x} [1]$

Step 1.1.3.3

Multiply $2$ by $3$ .

$6 x + \frac{d}{d x} [12 x] + \frac{d}{d x} [1]$

Step 1.1.4

Evaluate $\frac{d}{d x} [12 x]$ .

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Step 1.1.4.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x$ with respect to $x$ is $12 \frac{d}{d x} [x]$ .

$6 x + 12 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 1.1.4.2

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

$6 x + 12 \cdot 1 + \frac{d}{d x} [1]$

Step 1.1.4.3

Multiply $12$ by $1$ .

$6 x + 12 + \frac{d}{d x} [1]$

Step 1.1.5

Differentiate using the Constant Rule.

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Step 1.1.5.1

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

$6 x + 12 + 0$

Step 1.1.5.2

Add $6 x + 12$ and $0$ .

$6 x + 12$

Step 1.2

Substitute the lower limit in for $x$ in $u = 3 x^{2} + 12 x + 1$ .

$u_{lower} = 3 \cdot 0^{2} + 12 \cdot 0 + 1$

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 $0$ .

$u_{lower} = 3 \cdot 0 + 12 \cdot 0 + 1$

Step 1.3.1.2

Multiply $3$ by $0$ .

$u_{lower} = 0 + 12 \cdot 0 + 1$

Step 1.3.1.3

Multiply $12$ by $0$ .

$u_{lower} = 0 + 0 + 1$

Step 1.3.2

Add $0$ and $0$ .

$u_{lower} = 0 + 1$

Step 1.3.3

Add $0$ and $1$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = 3 x^{2} + 12 x + 1$ .

$u_{upper} = 3 \cdot 1^{2} + 12 \cdot 1 + 1$

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.

$u_{upper} = 3 \cdot 1 + 12 \cdot 1 + 1$

Step 1.5.1.2

Multiply $3$ by $1$ .

$u_{upper} = 3 + 12 \cdot 1 + 1$

Step 1.5.1.3

Multiply $12$ by $1$ .

$u_{upper} = 3 + 12 + 1$

Step 1.5.2

Add $3$ and $12$ .

$u_{upper} = 15 + 1$

Step 1.5.3

Add $15$ and $1$ .

$u_{upper} = 16$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 16$

Step 1.7

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

$\int_{1}^{16} u^{\frac{1}{2}} \frac{1}{6} d u$

Step 2

Combine $u^{\frac{1}{2}}$ and $\frac{1}{6}$ .

$\int_{1}^{16} \frac{u^{\frac{1}{2}}}{6} d u$

Step 3

Since $\frac{1}{6}$ is constant with respect to $u$ , move $\frac{1}{6}$ out of the integral.

$\frac{1}{6} \int_{1}^{16} u^{\frac{1}{2}} d u$

Step 4

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

$\frac{1}{6} \frac{2}{3} u^{\frac{3}{2}}]_{1}^{16}$

Step 5

Simplify the expression.

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Step 5.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $16$ and at $1$ .

$\frac{1}{6} ((\frac{2}{3} \cdot 16^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2

Simplify.

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Step 5.2.1

Rewrite $16$ as $4^{2}$ .

$\frac{1}{6} (\frac{2}{3} \cdot {(4^{2})}^{\frac{3}{2}} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2.2

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

$\frac{1}{6} (\frac{2}{3} \cdot 4^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2.3

Cancel the common factor of $2$ .

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Step 5.2.3.1

Cancel the common factor.

$\frac{1}{6} (\frac{2}{3} \cdot 4^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2.3.2

Rewrite the expression.

$\frac{1}{6} (\frac{2}{3} \cdot 4^{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.2.4

Raise $4$ to the power of $3$ .

$\frac{1}{6} (\frac{2}{3} \cdot 64 - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.3

Simplify.

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Step 5.3.1

Combine $\frac{2}{3}$ and $64$ .

$\frac{1}{6} (\frac{2 \cdot 64}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.3.2

Multiply $2$ by $64$ .

$\frac{1}{6} (\frac{128}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}})$

Step 5.4

Simplify the expression.

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Step 5.4.1

One to any power is one.

$\frac{1}{6} (\frac{128}{3} - \frac{2}{3} \cdot 1)$

Step 5.4.2

Multiply $- 1$ by $1$ .

$\frac{1}{6} (\frac{128}{3} - \frac{2}{3})$

Step 5.5

Simplify.

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Step 5.5.1

Combine the numerators over the common denominator.

$\frac{1}{6} \cdot \frac{128 - 2}{3}$

Step 5.5.2

Subtract $2$ from $128$ .

$\frac{1}{6} \cdot \frac{126}{3}$

Step 5.5.3

Cancel the common factor of $126$ and $3$ .

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Step 5.5.3.1

Factor $3$ out of $126$ .

$\frac{1}{6} \cdot \frac{3 \cdot 42}{3}$

Step 5.5.3.2

Cancel the common factors.

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Step 5.5.3.2.1

Factor $3$ out of $3$ .

$\frac{1}{6} \cdot \frac{3 \cdot 42}{3 (1)}$

Step 5.5.3.2.2

Cancel the common factor.

$\frac{1}{6} \cdot \frac{3 \cdot 42}{3 \cdot 1}$

Step 5.5.3.2.3

Rewrite the expression.

$\frac{1}{6} \cdot \frac{42}{1}$

Step 5.5.3.2.4

Divide $42$ by $1$ .

$\frac{1}{6} \cdot 42$

Step 5.6

Simplify.

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Step 5.6.1

Combine $\frac{1}{6}$ and $42$ .

$\frac{42}{6}$

Step 5.6.2

Cancel the common factor of $42$ and $6$ .

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Step 5.6.2.1

Factor $6$ out of $42$ .

$\frac{6 \cdot 7}{6}$

Step 5.6.2.2

Cancel the common factors.

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Step 5.6.2.2.1

Factor $6$ out of $6$ .

$\frac{6 \cdot 7}{6 (1)}$

Step 5.6.2.2.2

Cancel the common factor.

$\frac{6 \cdot 7}{6 \cdot 1}$

Step 5.6.2.2.3

Rewrite the expression.

$\frac{7}{1}$

Step 5.6.2.2.4

Divide $7$ by $1$ .

$7$