Calculus Examples

$\int_{0}^{1} 3000 (t + 5) d t$

Step 1

Multiply $3000 (t + 5)$ .

$\int_{0}^{1} 3000 t + 3000 \cdot 5 d t$

Step 2

Multiply $3000$ by $5$ .

$\int_{0}^{1} 3000 t + 15000 d t$

Step 3

Split the single integral into multiple integrals.

$\int_{0}^{1} 3000 t d t + \int_{0}^{1} 15000 d t$

Step 4

Since $3000$ is constant with respect to $t$ , move $3000$ out of the integral.

$3000 \int_{0}^{1} t d t + \int_{0}^{1} 15000 d t$

Step 5

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

$3000 (\frac{1}{2} t^{2}]_{0}^{1}) + \int_{0}^{1} 15000 d t$

Step 6

Combine $\frac{1}{2}$ and $t^{2}$ .

$3000 (\frac{t^{2}}{2}]_{0}^{1}) + \int_{0}^{1} 15000 d t$

Step 7

Apply the constant rule.

$3000 (\frac{t^{2}}{2}]_{0}^{1}) + 15000 t]_{0}^{1}$

Step 8

Substitute and simplify.

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Step 8.1

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

$3000 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) + 15000 t]_{0}^{1}$

Step 8.2

Evaluate $15000 t$ at $1$ and at $0$ .

$3000 (\frac{1^{2}}{2} - \frac{0^{2}}{2}) + 15000 \cdot 1 - 15000 \cdot 0$

Step 8.3

Simplify.

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Step 8.3.1

One to any power is one.

$3000 (\frac{1}{2} - \frac{0^{2}}{2}) + 15000 \cdot 1 - 15000 \cdot 0$

Step 8.3.2

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

$3000 (\frac{1}{2} - \frac{0}{2}) + 15000 \cdot 1 - 15000 \cdot 0$

Step 8.3.3

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

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Step 8.3.3.1

Factor $2$ out of $0$ .

$3000 (\frac{1}{2} - \frac{2 (0)}{2}) + 15000 \cdot 1 - 15000 \cdot 0$

Step 8.3.3.2

Cancel the common factors.

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Step 8.3.3.2.1

Factor $2$ out of $2$ .

$3000 (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + 15000 \cdot 1 - 15000 \cdot 0$

Step 8.3.3.2.2

Cancel the common factor.

$3000 (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1}) + 15000 \cdot 1 - 15000 \cdot 0$

Step 8.3.3.2.3

Rewrite the expression.

$3000 (\frac{1}{2} - \frac{0}{1}) + 15000 \cdot 1 - 15000 \cdot 0$

Step 8.3.3.2.4

Divide $0$ by $1$ .

$3000 (\frac{1}{2} - 0) + 15000 \cdot 1 - 15000 \cdot 0$

Step 8.3.4

Multiply $- 1$ by $0$ .

$3000 (\frac{1}{2} + 0) + 15000 \cdot 1 - 15000 \cdot 0$

Step 8.3.5

Add $\frac{1}{2}$ and $0$ .

$3000 (\frac{1}{2}) + 15000 \cdot 1 - 15000 \cdot 0$

Step 8.3.6

Combine $3000$ and $\frac{1}{2}$ .

$\frac{3000}{2} + 15000 \cdot 1 - 15000 \cdot 0$

Step 8.3.7

Cancel the common factor of $3000$ and $2$ .

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Step 8.3.7.1

Factor $2$ out of $3000$ .

$\frac{2 \cdot 1500}{2} + 15000 \cdot 1 - 15000 \cdot 0$

Step 8.3.7.2

Cancel the common factors.

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Step 8.3.7.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 1500}{2 (1)} + 15000 \cdot 1 - 15000 \cdot 0$

Step 8.3.7.2.2

Cancel the common factor.

$\frac{2 \cdot 1500}{2 \cdot 1} + 15000 \cdot 1 - 15000 \cdot 0$

Step 8.3.7.2.3

Rewrite the expression.

$\frac{1500}{1} + 15000 \cdot 1 - 15000 \cdot 0$

Step 8.3.7.2.4

Divide $1500$ by $1$ .

$1500 + 15000 \cdot 1 - 15000 \cdot 0$

Step 8.3.8

Multiply $15000$ by $1$ .

$1500 + 15000 - 15000 \cdot 0$

Step 8.3.9

Multiply $- 15000$ by $0$ .

$1500 + 15000 + 0$

Step 8.3.10

Add $15000$ and $0$ .

$1500 + 15000$

Step 8.3.11

Add $1500$ and $15000$ .

$16500$

Step 9