Calculus Examples

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

Step 1

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

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

Step 2

Substitute and simplify.

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Step 2.1

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

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

Step 2.2

Simplify.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Cancel the common factor of $2$ .

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Step 2.2.3.1

Cancel the common factor.

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

Step 2.2.3.2

Rewrite the expression.

$\frac{2}{7} \cdot 2^{7} - \frac{2}{7} \cdot 1^{\frac{7}{2}}$

Step 2.2.4

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

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

Step 2.2.5

Combine $\frac{2}{7}$ and $128$ .

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

Step 2.2.6

Multiply $2$ by $128$ .

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

Step 2.2.7

One to any power is one.

$\frac{256}{7} - \frac{2}{7} \cdot 1$

Step 2.2.8

Multiply $- 1$ by $1$ .

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

Step 2.2.9

Combine the numerators over the common denominator.

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

Step 2.2.10

Subtract $2$ from $256$ .

$\frac{254}{7}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{254}{7}$

Decimal Form:

$36.28571428 \dots$

Mixed Number Form:

$36 \frac{2}{7}$

Step 4