Calculus Examples

$\frac{x + 3}{\sqrt{x}}$

Step 1

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Move the negative in front of the fraction.

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

Step 4

Expand $(x + 3) x^{- \frac{1}{2}}$ .

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Step 4.1

Apply the distributive property.

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

Step 4.2

Raise $x$ to the power of $1$ .

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

Step 4.3

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

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

Step 4.4

Write $1$ as a fraction with a common denominator.

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Subtract $1$ from $2$ .

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

Step 4.7

Reorder $x^{\frac{1}{2}}$ and $3 x^{- \frac{1}{2}}$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

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

$3 (2 x^{\frac{1}{2}} + C) + \int x^{\frac{1}{2}} d x$

Step 8

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

$3 (2 x^{\frac{1}{2}} + C) + \frac{2}{3} x^{\frac{3}{2}} + C$

Step 9

Simplify.

$6 x^{\frac{1}{2}} + \frac{2}{3} x^{\frac{3}{2}} + C$