Calculus Examples

$\int_{1}^{4} 3 \sqrt{x} d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int_{1}^{4} \sqrt{x} d x$

Step 2

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

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

Step 3

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

$3 (\frac{2}{3} x^{\frac{3}{2}}]_{1}^{4})$

Step 4

Simplify the answer.

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Step 4.1

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

$3 (\frac{2 x^{\frac{3}{2}}}{3}]_{1}^{4})$

Step 4.2

Substitute and simplify.

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Step 4.2.1

Evaluate $\frac{2 x^{\frac{3}{2}}}{3}$ at $4$ and at $1$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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

Step 4.2.2.3

Cancel the common factor of $2$ .

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Step 4.2.2.3.1

Cancel the common factor.

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

Step 4.2.2.3.2

Rewrite the expression.

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

Step 4.2.2.4

Raise $2$ to the power of $3$ .

$3 (\frac{2 \cdot 8}{3} - \frac{2 \cdot 1^{\frac{3}{2}}}{3})$

Step 4.2.2.5

Multiply $2$ by $8$ .

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

Step 4.2.2.6

One to any power is one.

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

Step 4.2.2.7

Multiply $2$ by $1$ .

$3 (\frac{16}{3} - \frac{2}{3})$

Step 4.2.2.8

Combine the numerators over the common denominator.

$3 \frac{16 - 2}{3}$

Step 4.2.2.9

Subtract $2$ from $16$ .

$3 (\frac{14}{3})$

Step 4.2.2.10

Combine $3$ and $\frac{14}{3}$ .

$\frac{3 \cdot 14}{3}$

Step 4.2.2.11

Multiply $3$ by $14$ .

$\frac{42}{3}$

Step 4.2.2.12

Cancel the common factor of $42$ and $3$ .

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Step 4.2.2.12.1

Factor $3$ out of $42$ .

$\frac{3 \cdot 14}{3}$

Step 4.2.2.12.2

Cancel the common factors.

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Step 4.2.2.12.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 14}{3 (1)}$

Step 4.2.2.12.2.2

Cancel the common factor.

$\frac{3 \cdot 14}{3 \cdot 1}$

Step 4.2.2.12.2.3

Rewrite the expression.

$\frac{14}{1}$

Step 4.2.2.12.2.4

Divide $14$ by $1$ .

$14$

Step 5