Calculus Examples

$\int \frac{2 + \sqrt{x} + x}{x} d x$

Step 1

Split the fraction into multiple fractions.

$\int \frac{2}{x} + \frac{\sqrt{x}}{x} + \frac{x}{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int \frac{2}{x} d x + \int \frac{\sqrt{x}}{x} d x + \int \frac{x}{x} d x$

Step 3

Cancel the common factor of $x$ .

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Step 3.1

Cancel the common factor.

$\int \frac{2}{x} d x + \int \frac{\sqrt{x}}{x} d x + \int \frac{x}{x} d x$

Step 3.2

Rewrite the expression.

$\int \frac{2}{x} d x + \int \frac{\sqrt{x}}{x} d x + \int d x$

Step 4

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 5

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$2 (\ln (| x |) + C) + \int \frac{\sqrt{x}}{x} d x + \int d x$

Step 6

Simplify the expression.

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Step 6.1

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

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

Step 6.2

Simplify.

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Step 6.2.1

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 6.2.2

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

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Step 6.2.2.1

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

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Step 6.2.2.1.1

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

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

Step 6.2.2.1.2

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

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

Step 6.2.2.2

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

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

Step 6.2.2.3

Combine the numerators over the common denominator.

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

Step 6.2.2.4

Subtract $1$ from $2$ .

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

Step 6.3

Apply basic rules of exponents.

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Step 6.3.1

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

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

Step 6.3.2

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

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Step 6.3.2.1

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

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

Step 6.3.2.2

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

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

Step 6.3.2.3

Move the negative in front of the fraction.

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

Step 7

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

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

Step 8

Apply the constant rule.

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

Step 9

Simplify.

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