Calculus Examples

$p (t) = 9.2 t^{2} - \frac{2}{t}$

Step 1

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

$P (t) = \int p (t) d t$

Step 2

Set up the integral to solve.

$P (t) = \int 9.2 t^{2} - \frac{2}{t} d t$

Step 3

Split the single integral into multiple integrals.

$\int 9.2 t^{2} d t + \int - \frac{2}{t} d t$

Step 4

Since $9.2$ is constant with respect to $t$ , move $9.2$ out of the integral.

$9.2 \int t^{2} d t + \int - \frac{2}{t} d t$

Step 5

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

$9.2 (\frac{1}{3} t^{3} + C) + \int - \frac{2}{t} d t$

Step 6

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

$9.2 (\frac{1}{3} t^{3} + C) - \int \frac{2}{t} d t$

Step 7

Since $2$ is constant with respect to $t$ , move $2$ out of the integral.

$9.2 (\frac{1}{3} t^{3} + C) - (2 \int \frac{1}{t} d t)$

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{3}$ and $t^{3}$ .

$9.2 (\frac{t^{3}}{3} + C) - (2 \int \frac{1}{t} d t)$

Step 8.2

Multiply $2$ by $- 1$ .

$9.2 (\frac{t^{3}}{3} + C) - 2 \int \frac{1}{t} d t$

Step 9

The integral of $\frac{1}{t}$ with respect to $t$ is $\ln (| t |)$ .

$9.2 (\frac{t^{3}}{3} + C) - 2 (\ln (| t |) + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

$\frac{9.2 t^{3}}{3} - 2 \ln (| t |) + C$

Step 10.2

Reorder terms.

$\frac{9.2}{3} t^{3} - 2 \ln (| t |) + C$

Step 11

The answer is the antiderivative of the function $p (t) = 9.2 t^{2} - \frac{2}{t}$ .

$P (t) =$ $\frac{9.2}{3} t^{3} - 2 \ln (| t |) + C$