Calculus Examples

$\int_{0}^{3} t - 5^{t} d t$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{3} t d t + \int_{0}^{3} - 5^{t} d t$

Step 2

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

$\frac{1}{2} t^{2}]_{0}^{3} + \int_{0}^{3} - 5^{t} d t$

Step 3

Since $- 1$ is constant with respect to $t$ , move $- 1$ out of the integral.

$\frac{1}{2} t^{2}]_{0}^{3} - \int_{0}^{3} 5^{t} d t$

Step 4

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

$\frac{1}{2} t^{2}]_{0}^{3} - (\frac{5^{t}}{\ln (5)}]_{0}^{3})$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{1}{2} t^{2}$ at $3$ and at $0$ .

$(\frac{1}{2} \cdot 3^{2}) - \frac{1}{2} \cdot 0^{2} - (\frac{5^{t}}{\ln (5)}]_{0}^{3})$

Step 5.2

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

$\frac{1}{2} \cdot 3^{2} - \frac{1}{2} \cdot 0^{2} - (\frac{5^{3}}{\ln (5)} - \frac{5^{0}}{\ln (5)})$

Step 5.3

Simplify.

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Step 5.3.1

Raise $3$ to the power of $2$ .

$\frac{1}{2} \cdot 9 - \frac{1}{2} \cdot 0^{2} - (\frac{5^{3}}{\ln (5)} - \frac{5^{0}}{\ln (5)})$

Step 5.3.2

Combine $\frac{1}{2}$ and $9$ .

$\frac{9}{2} - \frac{1}{2} \cdot 0^{2} - (\frac{5^{3}}{\ln (5)} - \frac{5^{0}}{\ln (5)})$

Step 5.3.3

Raising $0$ to any positive power yields $0$ .

$\frac{9}{2} - \frac{1}{2} \cdot 0 - (\frac{5^{3}}{\ln (5)} - \frac{5^{0}}{\ln (5)})$

Step 5.3.4

Multiply $0$ by $- 1$ .

$\frac{9}{2} + 0 (\frac{1}{2}) - (\frac{5^{3}}{\ln (5)} - \frac{5^{0}}{\ln (5)})$

Step 5.3.5

Multiply $0$ by $\frac{1}{2}$ .

$\frac{9}{2} + 0 - (\frac{5^{3}}{\ln (5)} - \frac{5^{0}}{\ln (5)})$

Step 5.3.6

Add $\frac{9}{2}$ and $0$ .

$\frac{9}{2} - (\frac{5^{3}}{\ln (5)} - \frac{5^{0}}{\ln (5)})$

Step 5.3.7

Raise $5$ to the power of $3$ .

$\frac{9}{2} - (\frac{125}{\ln (5)} - \frac{5^{0}}{\ln (5)})$

Step 5.3.8

Anything raised to $0$ is $1$ .

$\frac{9}{2} - (\frac{125}{\ln (5)} - \frac{1}{\ln (5)})$

Step 5.3.9

Combine the numerators over the common denominator.

$\frac{9}{2} - \frac{125 - 1}{\ln (5)}$

Step 5.3.10

Subtract $1$ from $125$ .

$\frac{9}{2} - \frac{124}{\ln (5)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{9}{2} - \frac{124}{\ln (5)}$

Decimal Form:

$- 72.54553188 \dots$

Step 7