Calculus Examples

$\int_{1}^{4} \sqrt{16 x^{5}} d x$

Step 1

Simplify.

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Step 1.1

Rewrite $16 x^{5}$ as ${(4 x^{2})}^{2} x$ .

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Step 1.1.1

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

$\int_{1}^{4} \sqrt{4^{2} x^{5}} d x$

Step 1.1.2

Factor out $x^{4}$ .

$\int_{1}^{4} \sqrt{4^{2} (x^{4} x)} d x$

Step 1.1.3

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\int_{1}^{4} \sqrt{4^{2} ({(x^{2})}^{2} x)} d x$

Step 1.1.4

Rewrite $4^{2} {(x^{2})}^{2}$ as ${(4 x^{2})}^{2}$ .

$\int_{1}^{4} \sqrt{{(4 x^{2})}^{2} x} d x$

Step 1.2

Pull terms out from under the radical.

$\int_{1}^{4} 4 x^{2} \sqrt{x} d x$

Step 2

Simplify.

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Step 2.1

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

$\int_{1}^{4} 4 x^{2} x^{\frac{1}{2}} d x$

Step 2.2

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

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Step 2.2.1

Move $x^{\frac{1}{2}}$ .

$\int_{1}^{4} 4 (x^{\frac{1}{2}} x^{2}) d x$

Step 2.2.2

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

$\int_{1}^{4} 4 x^{\frac{1}{2} + 2} d x$

Step 2.2.3

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

$\int_{1}^{4} 4 x^{\frac{1}{2} + 2 \cdot \frac{2}{2}} d x$

Step 2.2.4

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

$\int_{1}^{4} 4 x^{\frac{1}{2} + \frac{2 \cdot 2}{2}} d x$

Step 2.2.5

Combine the numerators over the common denominator.

$\int_{1}^{4} 4 x^{\frac{1 + 2 \cdot 2}{2}} d x$

Step 2.2.6

Simplify the numerator.

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Step 2.2.6.1

Multiply $2$ by $2$ .

$\int_{1}^{4} 4 x^{\frac{1 + 4}{2}} d x$

Step 2.2.6.2

Add $1$ and $4$ .

$\int_{1}^{4} 4 x^{\frac{5}{2}} d x$

Step 3

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int_{1}^{4} x^{\frac{5}{2}} d x$

Step 4

By the Power Rule, the integral of $x^{\frac{5}{2}}$ with respect to $x$ is $\frac{2}{7} x^{\frac{7}{2}}$ .

$4 (\frac{2}{7} x^{\frac{7}{2}}]_{1}^{4})$

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{2}{7}$ and $x^{\frac{7}{2}}$ .

$4 (\frac{2 x^{\frac{7}{2}}}{7}]_{1}^{4})$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $\frac{2 x^{\frac{7}{2}}}{7}$ at $4$ and at $1$ .

$4 ((\frac{2 \cdot 4^{\frac{7}{2}}}{7}) - \frac{2 \cdot 1^{\frac{7}{2}}}{7})$

Step 5.2.2

Simplify.

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Step 5.2.2.1

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

$4 (\frac{2 \cdot {(2^{2})}^{\frac{7}{2}}}{7} - \frac{2 \cdot 1^{\frac{7}{2}}}{7})$

Step 5.2.2.2

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

$4 (\frac{2 \cdot 2^{2 (\frac{7}{2})}}{7} - \frac{2 \cdot 1^{\frac{7}{2}}}{7})$

Step 5.2.2.3

Cancel the common factor of $2$ .

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Step 5.2.2.3.1

Cancel the common factor.

$4 (\frac{2 \cdot 2^{2 (\frac{7}{2})}}{7} - \frac{2 \cdot 1^{\frac{7}{2}}}{7})$

Step 5.2.2.3.2

Rewrite the expression.

$4 (\frac{2 \cdot 2^{7}}{7} - \frac{2 \cdot 1^{\frac{7}{2}}}{7})$

Step 5.2.2.4

Raise $2$ to the power of $7$ .

$4 (\frac{2 \cdot 128}{7} - \frac{2 \cdot 1^{\frac{7}{2}}}{7})$

Step 5.2.2.5

Multiply $2$ by $128$ .

$4 (\frac{256}{7} - \frac{2 \cdot 1^{\frac{7}{2}}}{7})$

Step 5.2.2.6

One to any power is one.

$4 (\frac{256}{7} - \frac{2 \cdot 1}{7})$

Step 5.2.2.7

Multiply $2$ by $1$ .

$4 (\frac{256}{7} - \frac{2}{7})$

Step 5.2.2.8

Combine the numerators over the common denominator.

$4 \frac{256 - 2}{7}$

Step 5.2.2.9

Subtract $2$ from $256$ .

$4 (\frac{254}{7})$

Step 5.2.2.10

Combine $4$ and $\frac{254}{7}$ .

$\frac{4 \cdot 254}{7}$

Step 5.2.2.11

Multiply $4$ by $254$ .

$\frac{1016}{7}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{1016}{7}$

Decimal Form:

$145.14285714 \dots$

Mixed Number Form:

$145 \frac{1}{7}$

Step 7