Calculus Examples

$v (t) = 30 (25 - t^{2})$

Step 1

The function $V (t)$ can be found by finding the indefinite integral of the derivative $v (t)$ .

$V (t) = \int v (t) d t$

Step 2

Set up the integral to solve.

$V (t) = \int 30 (25 - t^{2}) d t$

Step 3

Multiply $30 (25 - t^{2})$ .

$\int 30 \cdot 25 + 30 (- t^{2}) d t$

Step 4

Simplify.

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Step 4.1

Multiply $30$ by $25$ .

$\int 750 + 30 (- t^{2}) d t$

Step 4.2

Multiply $- 1$ by $30$ .

$\int 750 - 30 t^{2} d t$

Step 5

Split the single integral into multiple integrals.

$\int 750 d t + \int - 30 t^{2} d t$

Step 6

Apply the constant rule.

$750 t + C + \int - 30 t^{2} d t$

Step 7

Since $- 30$ is constant with respect to $t$ , move $- 30$ out of the integral.

$750 t + C - 30 \int t^{2} d t$

Step 8

By the Power Rule, the integral of $t^{2}$ with respect to $t$ is $\frac{1}{3} t^{3}$ .

$750 t + C - 30 (\frac{1}{3} t^{3} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

$750 t - 30 (\frac{1}{3}) t^{3} + C$

Step 9.2

Simplify.

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Step 9.2.1

Combine $- 30$ and $\frac{1}{3}$ .

$750 t + \frac{- 30}{3} t^{3} + C$

Step 9.2.2

Cancel the common factor of $- 30$ and $3$ .

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Step 9.2.2.1

Factor $3$ out of $- 30$ .

$750 t + \frac{3 \cdot - 10}{3} t^{3} + C$

Step 9.2.2.2

Cancel the common factors.

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Step 9.2.2.2.1

Factor $3$ out of $3$ .

$750 t + \frac{3 \cdot - 10}{3 (1)} t^{3} + C$

Step 9.2.2.2.2

Cancel the common factor.

$750 t + \frac{3 \cdot - 10}{3 \cdot 1} t^{3} + C$

Step 9.2.2.2.3

Rewrite the expression.

$750 t + \frac{- 10}{1} t^{3} + C$

Step 9.2.2.2.4

Divide $- 10$ by $1$ .

$750 t - 10 t^{3} + C$

Step 10

The answer is the antiderivative of the function $v (t) = 30 (25 - t^{2})$ .

$V (t) =$ $750 t - 10 t^{3} + C$