Calculus Examples

$\frac{(x^{2} + 1)}{\sqrt{x}}$

Step 1

Write $\frac{(x^{2} + 1)}{\sqrt{x}}$ as a function.

$f (x) = \frac{(x^{2} + 1)}{\sqrt{x}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{x^{2} + 1}{\sqrt{x}} d x$

Step 4

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

$\int \frac{x^{2} + 1}{x^{\frac{1}{2}}} d x$

Step 5

Move $x^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\int (x^{2} + 1) {(x^{\frac{1}{2}})}^{- 1} d x$

Step 6

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 1}$ .

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Step 6.1

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

$\int (x^{2} + 1) x^{\frac{1}{2} \cdot - 1} d x$

Step 6.2

Combine $\frac{1}{2}$ and $- 1$ .

$\int (x^{2} + 1) x^{\frac{- 1}{2}} d x$

Step 6.3

Move the negative in front of the fraction.

$\int (x^{2} + 1) x^{- \frac{1}{2}} d x$

Step 7

Expand $(x^{2} + 1) x^{- \frac{1}{2}}$ .

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Step 7.1

Apply the distributive property.

$\int x^{2} x^{- \frac{1}{2}} + 1 x^{- \frac{1}{2}} d x$

Step 7.2

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

$\int x^{2 - \frac{1}{2}} + 1 x^{- \frac{1}{2}} d x$

Step 7.3

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

$\int x^{2 \cdot \frac{2}{2} - \frac{1}{2}} + 1 x^{- \frac{1}{2}} d x$

Step 7.4

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

$\int x^{\frac{2 \cdot 2}{2} - \frac{1}{2}} + 1 x^{- \frac{1}{2}} d x$

Step 7.5

Combine the numerators over the common denominator.

$\int x^{\frac{2 \cdot 2 - 1}{2}} + 1 x^{- \frac{1}{2}} d x$

Step 7.6

Simplify the numerator.

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Step 7.6.1

Multiply $2$ by $2$ .

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

Step 7.6.2

Subtract $1$ from $4$ .

$\int x^{\frac{3}{2}} + 1 x^{- \frac{1}{2}} d x$

Step 7.7

Multiply $x^{- \frac{1}{2}}$ by $1$ .

$\int x^{\frac{3}{2}} + x^{- \frac{1}{2}} d x$

Step 8

Split the single integral into multiple integrals.

$\int x^{\frac{3}{2}} d x + \int x^{- \frac{1}{2}} d x$

Step 9

By the Power Rule, the integral of $x^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{5} x^{\frac{5}{2}}$ .

$\frac{2}{5} x^{\frac{5}{2}} + C + \int x^{- \frac{1}{2}} d x$

Step 10

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

$\frac{2}{5} x^{\frac{5}{2}} + C + 2 x^{\frac{1}{2}} + C$

Step 11

Simplify.

$\frac{2}{5} x^{\frac{5}{2}} + 2 x^{\frac{1}{2}} + C$

Step 12

The answer is the antiderivative of the function $f (x) = \frac{x^{2} + 1}{\sqrt{x}}$ .

$F (x) =$ $\frac{2}{5} x^{\frac{5}{2}} + 2 x^{\frac{1}{2}} + C$