Find the Antiderivative p(t)=9.2t^2-2/t

Solution:

Step 1

Identify the antiderivative $P(t)$ by integrating the given function $p(t)$. $$P(t)=\int p(t)dt$$

Step 2

Write down the integral that needs to be solved.

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

Step 3

Decompose the integral into a sum of integrals.

$$\int 9.2t^{2}dt-\int \frac{2}{t}dt$$

Step 4

Extract the constant $9.2$ from the integral as it is not dependent on $t$.

$$9.2\int t^{2}dt-\int \frac{2}{t}dt$$

Step 5

Apply the Power Rule for integration to find the integral of $t^2$.

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

Step 6

Factor out the constant $-2$ from the second integral.

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

Step 7

Isolate the constant $2$ from the integral.

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

Step 8

Proceed to simplify the expression.

Step 8.1

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

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

Step 8.2

Multiply $2$ by the integral of $\frac{1}{t}$.

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

Step 9

Integrate $\frac{1}{t}$ to get the natural logarithm.

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

Step 10

Simplify the expression further.

Step 10.1

Combine terms to get the simplified form.

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

Step 10.2

Rearrange the terms if necessary.

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

Step 11

Conclude with the antiderivative of $p(t)=9.2t^2-\frac{2}{t}$.

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

Knowledge Notes:

1. Indefinite Integral: The antiderivative or indefinite integral is the reverse process of differentiation. It is represented by the integral sign followed by the function and the differential, e.g., $\int f(x)dx$.

2. Power Rule for Integration: This rule states that the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1}$, provided $n\ne -1$. For example, $\int t^{2}dt=\frac{t^3}{3}+C$.

3. Integral of Reciprocal: The integral of $\frac{1}{x}$ with respect to $x$ is $\ln |x|+C$, where $C$ is the constant of integration.

4. Constant Factor Rule: A constant factor can be factored out of an integral. If $k$ is a constant and $f(x)$ is a function, then $\int kf(x)dx=k\int f(x)dx$.

5. Natural Logarithm: The natural logarithm, denoted as $\ln (x)$, is the inverse function of the exponential function $e^x$. The integral of $\frac{1}{x}$ is $\ln |x|$.

6. Absolute Value: In integrals involving $\frac{1}{x}$, the absolute value is used to ensure the logarithm is defined for all real numbers except zero, as $\ln (x)$ is undefined for $x\le 0$.