Calculus Examples

$\int_{0}^{1} 5^{\frac{3 x}{2}} d x$

Step 1

Let $u = \frac{3 x}{2}$ . Then $d u = \frac{3}{2} d x$ , so $\frac{2}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = \frac{3 x}{2}$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $\frac{3 x}{2}$ .

$\frac{d}{d x} [\frac{3 x}{2}]$

Step 1.1.2

Since $\frac{3}{2}$ is constant with respect to $x$ , the derivative of $\frac{3 x}{2}$ with respect to $x$ is $\frac{3}{2} \frac{d}{d x} [x]$ .

$\frac{3}{2} \frac{d}{d x} [x]$

Step 1.1.3

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

$\frac{3}{2} \cdot 1$

Step 1.1.4

Multiply $\frac{3}{2}$ by $1$ .

$\frac{3}{2}$

Step 1.2

Substitute the lower limit in for $x$ in $u = \frac{3 x}{2}$ .

$u_{lower} = \frac{3 \cdot 0}{2}$

Step 1.3

Simplify.

Tap for more steps...

Step 1.3.1

Cancel the common factor of $0$ and $2$ .

Tap for more steps...

Step 1.3.1.1

Factor $2$ out of $3 \cdot 0$ .

$u_{lower} = \frac{2 (3 \cdot 0)}{2}$

Step 1.3.1.2

Cancel the common factors.

Tap for more steps...

Step 1.3.1.2.1

Factor $2$ out of $2$ .

$u_{lower} = \frac{2 (3 \cdot 0)}{2 (1)}$

Step 1.3.1.2.2

Cancel the common factor.

$u_{lower} = \frac{2 (3 \cdot 0)}{2 \cdot 1}$

Step 1.3.1.2.3

Rewrite the expression.

$u_{lower} = \frac{3 \cdot 0}{1}$

Step 1.3.1.2.4

Divide $3 \cdot 0$ by $1$ .

$u_{lower} = 3 \cdot 0$

Step 1.3.2

Multiply $3$ by $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u = \frac{3 x}{2}$ .

$u_{upper} = \frac{3 \cdot 1}{2}$

Step 1.5

Multiply $3$ by $1$ .

$u_{upper} = \frac{3}{2}$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = \frac{3}{2}$

Step 1.7

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

$\int_{0}^{\frac{3}{2}} 5^{u} \cdot \frac{1}{\frac{3}{2}} d u$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Multiply by the reciprocal of the fraction to divide by $\frac{3}{2}$ .

$\int_{0}^{\frac{3}{2}} 5^{u} \cdot (1 (\frac{2}{3})) d u$

Step 2.2

Multiply $\frac{2}{3}$ by $1$ .

$\int_{0}^{\frac{3}{2}} 5^{u} \cdot \frac{2}{3} d u$

Step 2.3

Combine $5^{u}$ and $\frac{2}{3}$ .

$\int_{0}^{\frac{3}{2}} \frac{5^{u} \cdot 2}{3} d u$

Step 2.4

Move $2$ to the left of $5^{u}$ .

$\int_{0}^{\frac{3}{2}} \frac{2 \cdot 5^{u}}{3} d u$

Step 3

Since $\frac{2}{3}$ is constant with respect to $u$ , move $\frac{2}{3}$ out of the integral.

$\frac{2}{3} \int_{0}^{\frac{3}{2}} 5^{u} d u$

Step 4

The integral of $5^{u}$ with respect to $u$ is $\frac{5^{u}}{\ln (5)}$ .

$\frac{2}{3} \frac{5^{u}}{\ln (5)}]_{0}^{\frac{3}{2}}$

Step 5

Combine $\frac{2}{3}$ and $\frac{5^{u}}{\ln (5)}]_{0}^{\frac{3}{2}}$ .

$\frac{2 (\frac{5^{u}}{\ln (5)}]_{0}^{\frac{3}{2}})}{3}$

Step 6

Substitute and simplify.

Tap for more steps...

Step 6.1

Evaluate $\frac{5^{u}}{\ln (5)}$ at $\frac{3}{2}$ and at $0$ .

$\frac{2 ((\frac{5^{\frac{3}{2}}}{\ln (5)}) - \frac{5^{0}}{\ln (5)})}{3}$

Step 6.2

Anything raised to $0$ is $1$ .

$\frac{2 (\frac{5^{\frac{3}{2}}}{\ln (5)} - \frac{1}{\ln (5)})}{3}$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Combine the numerators over the common denominator.

$\frac{2 \frac{5^{\frac{3}{2}} - 1}{\ln (5)}}{3}$

Step 7.2

Combine $2$ and $\frac{5^{\frac{3}{2}} - 1}{\ln (5)}$ .

$\frac{\frac{2 (5^{\frac{3}{2}} - 1)}{\ln (5)}}{3}$

Step 7.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 (5^{\frac{3}{2}} - 1)}{\ln (5)} \cdot \frac{1}{3}$

Step 7.4

Combine.

$\frac{2 (5^{\frac{3}{2}} - 1) \cdot 1}{\ln (5) \cdot 3}$

Step 7.5

Multiply $2$ by $1$ .

$\frac{2 (5^{\frac{3}{2}} - 1)}{\ln (5) \cdot 3}$

Step 7.6

Move $3$ to the left of $\ln (5)$ .

$\frac{2 (5^{\frac{3}{2}} - 1)}{3 \ln (5)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{2 (5^{\frac{3}{2}} - 1)}{3 \ln (5)}$

Decimal Form:

$4.21693387 \dots$

Step 9