Calculus Examples

$\int e^{\ln (x)} \frac{d x}{x} d x$

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

Simplify.

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Step 2.1

Exponentiation and log are inverse functions.

$\int x \frac{d x}{x} d x$

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

$\int x \frac{d x}{x} d x$

Step 2.2.2

Divide $d$ by $1$ .

$\int x d d x$

Step 3

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

$d \int x d x$

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

Rewrite $d (\frac{1}{2} x^{2} + C)$ as $d \frac{1}{2} x^{2} + C$ .

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

Step 5.2

Simplify.

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Step 5.2.1

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

$\frac{d}{2} x^{2} + C$

Step 5.2.2

Combine $\frac{d}{2}$ and $x^{2}$ .

$\frac{d x^{2}}{2} + C$

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