Calculus Examples

$\int_{1}^{e^{x}} \ln (t) d t$

Step 1

Take the derivative of $\int_{1}^{e^{x}} \ln (t) d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

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

Step 2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$e^{x} \ln (e^{x})$

Step 3

Use logarithm rules to move $x$ out of the exponent.

$e^{x} (x \ln (e))$

Step 4

The natural logarithm of $e$ is $1$ .

$e^{x} (x \cdot 1)$

Step 5

Multiply $x$ by $1$ .

$e^{x} x$

Step 6

Reorder factors in $e^{x} x$ .

$x e^{x}$