Calculus Examples

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

Step 1

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

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Step 1.1

Simplify the expression.

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Step 1.1.1

Multiply $x^{4}$ by $1$ .

$\int_{2}^{4} \sqrt{x^{4}} d x + \int_{2}^{4} \sqrt{3} d x$

Step 1.1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{4}}$ as $x^{\frac{4}{2}}$ .

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

Step 1.1.3

Divide $4$ by $2$ .

$\int_{2}^{4} x^{2} d x + \int_{2}^{4} \sqrt{3} d x$

Step 1.2

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

$\frac{1}{3} x^{3}]_{2}^{4} + \int_{2}^{4} \sqrt{3} d x$

Step 1.3

Substitute and simplify.

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Step 1.3.1

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

$(\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} \cdot 2^{3} + \int_{2}^{4} \sqrt{3} d x$

Step 1.3.2

Simplify.

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Step 1.3.2.1

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

$\frac{1}{3} \cdot 64 - \frac{1}{3} \cdot 2^{3} + \int_{2}^{4} \sqrt{3} d x$

Step 1.3.2.2

Combine $\frac{1}{3}$ and $64$ .

$\frac{64}{3} - \frac{1}{3} \cdot 2^{3} + \int_{2}^{4} \sqrt{3} d x$

Step 1.3.2.3

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

$\frac{64}{3} - \frac{1}{3} \cdot 8 + \int_{2}^{4} \sqrt{3} d x$

Step 1.3.2.4

Multiply $8$ by $- 1$ .

$\frac{64}{3} - 8 (\frac{1}{3}) + \int_{2}^{4} \sqrt{3} d x$

Step 1.3.2.5

Combine $- 8$ and $\frac{1}{3}$ .

$\frac{64}{3} + \frac{- 8}{3} + \int_{2}^{4} \sqrt{3} d x$

Step 1.3.2.6

Move the negative in front of the fraction.

$\frac{64}{3} - \frac{8}{3} + \int_{2}^{4} \sqrt{3} d x$

Step 1.3.2.7

Combine the numerators over the common denominator.

$\frac{64 - 8}{3} + \int_{2}^{4} \sqrt{3} d x$

Step 1.3.2.8

Subtract $8$ from $64$ .

$\frac{56}{3} + \int_{2}^{4} \sqrt{3} d x$

Step 2

Evaluate $\int_{2}^{4} \sqrt{3} d x$ .

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Step 2.1

Apply the constant rule.

$\frac{56}{3} + \sqrt{3} x]_{2}^{4}$

Step 2.2

Substitute and simplify.

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Step 2.2.1

Evaluate $\sqrt{3} x$ at $4$ and at $2$ .

$\frac{56}{3} + (\sqrt{3} \cdot 4) - \sqrt{3} \cdot 2$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Move $4$ to the left of $\sqrt{3}$ .

$\frac{56}{3} + 4 \cdot \sqrt{3} - \sqrt{3} \cdot 2$

Step 2.2.2.2

Multiply $2$ by $- 1$ .

$\frac{56}{3} + 4 \sqrt{3} - 2 \sqrt{3}$

Step 2.2.2.3

Subtract $2 \sqrt{3}$ from $4 \sqrt{3}$ .

$\frac{56}{3} + 2 \sqrt{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{56}{3} + 2 \sqrt{3}$

Decimal Form:

$22.13076828 \dots$