Calculus Examples

$\int_{2}^{5} 355 t + 1125 d t$ udu

Step 1

Split the single integral into multiple integrals.

$\int_{2}^{5} 355 t d t + \int_{2}^{5} 1125 d t$

Step 2

Since $355$ is constant with respect to $t$ , move $355$ out of the integral.

$355 \int_{2}^{5} t d t + \int_{2}^{5} 1125 d t$

Step 3

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

$355 (\frac{1}{2} t^{2}]_{2}^{5}) + \int_{2}^{5} 1125 d t$

Step 4

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

$355 (\frac{t^{2}}{2}]_{2}^{5}) + \int_{2}^{5} 1125 d t$

Step 5

Apply the constant rule.

$355 (\frac{t^{2}}{2}]_{2}^{5}) + 1125 t]_{2}^{5}$

Step 6

Substitute and simplify.

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Step 6.1

Evaluate $\frac{t^{2}}{2}$ at $5$ and at $2$ .

$355 ((\frac{5^{2}}{2}) - \frac{2^{2}}{2}) + 1125 t]_{2}^{5}$

Step 6.2

Evaluate $1125 t$ at $5$ and at $2$ .

$355 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + 1125 \cdot 5 - 1125 \cdot 2$

Step 6.3

Simplify.

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Step 6.3.1

Raise $5$ to the power of $2$ .

$355 (\frac{25}{2} - \frac{2^{2}}{2}) + 1125 \cdot 5 - 1125 \cdot 2$

Step 6.3.2

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

$355 (\frac{25}{2} - \frac{4}{2}) + 1125 \cdot 5 - 1125 \cdot 2$

Step 6.3.3

Cancel the common factor of $4$ and $2$ .

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Step 6.3.3.1

Factor $2$ out of $4$ .

$355 (\frac{25}{2} - \frac{2 \cdot 2}{2}) + 1125 \cdot 5 - 1125 \cdot 2$

Step 6.3.3.2

Cancel the common factors.

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Step 6.3.3.2.1

Factor $2$ out of $2$ .

$355 (\frac{25}{2} - \frac{2 \cdot 2}{2 (1)}) + 1125 \cdot 5 - 1125 \cdot 2$

Step 6.3.3.2.2

Cancel the common factor.

$355 (\frac{25}{2} - \frac{2 \cdot 2}{2 \cdot 1}) + 1125 \cdot 5 - 1125 \cdot 2$

Step 6.3.3.2.3

Rewrite the expression.

$355 (\frac{25}{2} - \frac{2}{1}) + 1125 \cdot 5 - 1125 \cdot 2$

Step 6.3.3.2.4

Divide $2$ by $1$ .

$355 (\frac{25}{2} - 1 \cdot 2) + 1125 \cdot 5 - 1125 \cdot 2$

Step 6.3.4

Multiply $- 1$ by $2$ .

$355 (\frac{25}{2} - 2) + 1125 \cdot 5 - 1125 \cdot 2$

Step 6.3.5

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$355 (\frac{25}{2} - 2 \cdot \frac{2}{2}) + 1125 \cdot 5 - 1125 \cdot 2$

Step 6.3.6

Combine $- 2$ and $\frac{2}{2}$ .

$355 (\frac{25}{2} + \frac{- 2 \cdot 2}{2}) + 1125 \cdot 5 - 1125 \cdot 2$

Step 6.3.7

Combine the numerators over the common denominator.

$355 \frac{25 - 2 \cdot 2}{2} + 1125 \cdot 5 - 1125 \cdot 2$

Step 6.3.8

Simplify the numerator.

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Step 6.3.8.1

Multiply $- 2$ by $2$ .

$355 \frac{25 - 4}{2} + 1125 \cdot 5 - 1125 \cdot 2$

Step 6.3.8.2

Subtract $4$ from $25$ .

$355 (\frac{21}{2}) + 1125 \cdot 5 - 1125 \cdot 2$

Step 6.3.9

Combine $355$ and $\frac{21}{2}$ .

$\frac{355 \cdot 21}{2} + 1125 \cdot 5 - 1125 \cdot 2$

Step 6.3.10

Multiply $355$ by $21$ .

$\frac{7455}{2} + 1125 \cdot 5 - 1125 \cdot 2$

Step 6.3.11

Multiply $1125$ by $5$ .

$\frac{7455}{2} + 5625 - 1125 \cdot 2$

Step 6.3.12

Multiply $- 1125$ by $2$ .

$\frac{7455}{2} + 5625 - 2250$

Step 6.3.13

Subtract $2250$ from $5625$ .

$\frac{7455}{2} + 3375$

Step 6.3.14

To write $3375$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{7455}{2} + 3375 \cdot \frac{2}{2}$

Step 6.3.15

Combine $3375$ and $\frac{2}{2}$ .

$\frac{7455}{2} + \frac{3375 \cdot 2}{2}$

Step 6.3.16

Combine the numerators over the common denominator.

$\frac{7455 + 3375 \cdot 2}{2}$

Step 6.3.17

Simplify the numerator.

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Step 6.3.17.1

Multiply $3375$ by $2$ .

$\frac{7455 + 6750}{2}$

Step 6.3.17.2

Add $7455$ and $6750$ .

$\frac{14205}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{14205}{2}$

Decimal Form:

$7102.5$

Mixed Number Form:

$7102 \frac{1}{2}$

Step 8