Calculus Examples

$\int_{\ln (27)}^{\ln (216)} e^{\frac{x}{3}} d x$

Step 1

Let $u = \frac{x}{3}$ . Then $d u = \frac{1}{3} d x$ , so $3 d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = \frac{x}{3}$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $\frac{x}{3}$ .

$\frac{d}{d x} [\frac{x}{3}]$

Step 1.1.2

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x]$ .

$\frac{1}{3} \frac{d}{d x} [x]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{1}{3} \cdot 1$

Step 1.1.4

Multiply $\frac{1}{3}$ by $1$ .

$\frac{1}{3}$

Step 1.2

Substitute the lower limit in for $x$ in $u = \frac{x}{3}$ .

$u_{lower} = \frac{\ln (27)}{3}$

Step 1.3

Simplify.

Tap for more steps...

Step 1.3.1

Rewrite $\ln (27)$ as $\ln (3^{3})$ .

$u_{lower} = \frac{\ln (3^{3})}{3}$

Step 1.3.2

Expand $\ln (3^{3})$ by moving $3$ outside the logarithm.

$u_{lower} = \frac{3 \ln (3)}{3}$

Step 1.3.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.3.3.1

Cancel the common factor.

$u_{lower} = \frac{3 \ln (3)}{3}$

Step 1.3.3.2

Divide $\ln (3)$ by $1$ .

$u_{lower} = \ln (3)$

Step 1.4

Substitute the upper limit in for $x$ in $u = \frac{x}{3}$ .

$u_{upper} = \frac{\ln (216)}{3}$

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Rewrite $\frac{\ln (216)}{3}$ as $\frac{1}{3} \ln (216)$ .

$u_{upper} = \frac{1}{3} \ln (216)$

Step 1.5.2

Simplify $\frac{1}{3} \ln (216)$ by moving $\frac{1}{3}$ inside the logarithm.

$u_{upper} = \ln (216^{\frac{1}{3}})$

Step 1.5.3

Rewrite $216$ as $6^{3}$ .

$u_{upper} = \ln ({(6^{3})}^{\frac{1}{3}})$

Step 1.5.4

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

$u_{upper} = \ln (6^{3 (\frac{1}{3})})$

Step 1.5.5

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.5.5.1

Cancel the common factor.

$u_{upper} = \ln (6^{3 (\frac{1}{3})})$

Step 1.5.5.2

Rewrite the expression.

$u_{upper} = \ln (6^{1})$

Step 1.5.6

Evaluate the exponent.

$u_{upper} = \ln (6)$

Step 1.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = \ln (3)$

$u_{upper} = \ln (6)$

Step 1.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{\ln (3)}^{\ln (6)} e^{u} \frac{1}{\frac{1}{3}} d u$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{3}$ .

$\int_{\ln (3)}^{\ln (6)} e^{u} (1 \cdot 3) d u$

Step 2.2

Multiply $3$ by $1$ .

$\int_{\ln (3)}^{\ln (6)} e^{u} \cdot 3 d u$

Step 2.3

Move $3$ to the left of $e^{u}$ .

$\int_{\ln (3)}^{\ln (6)} 3 e^{u} d u$

Step 3

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

$3 \int_{\ln (3)}^{\ln (6)} e^{u} d u$

Step 4

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$3 (e^{u}]_{\ln (3)}^{\ln (6)})$

Step 5

Substitute and simplify.

Tap for more steps...

Step 5.1

Evaluate $e^{u}$ at $\ln (6)$ and at $\ln (3)$ .

$3 ((e^{\ln (6)}) - e^{\ln (3)})$

Step 5.2

Simplify.

Tap for more steps...

Step 5.2.1

Exponentiation and log are inverse functions.

$3 (6 - e^{\ln (3)})$

Step 5.2.2

Exponentiation and log are inverse functions.

$3 (6 - 1 \cdot 3)$

Step 5.2.3

Multiply $- 1$ by $3$ .

$3 (6 - 3)$

Step 5.2.4

Subtract $3$ from $6$ .

$3 \cdot 3$

Step 5.2.5

Multiply $3$ by $3$ .

$9$

Step 6