Find the Antiderivative p(t)=9.2t^2-2/t
The question is asking for the calculation of an antiderivative (also known as an indefinite integral) of the provided function p(t) with respect to the variable t. The function p(t) is given as a polynomial expression 9.2t^2 minus a rational expression 2/t, and you are expected to integrate this function term by term to find a new function P(t) such that the derivative of P(t) with respect to t equals the original function p(t). The result will also include an arbitrary constant of integration, typically denoted as C, because the antiderivative of a function is not unique – each constant added yields another valid antiderivative.
$p \left(\right. t \left.\right) = 9.2 t^{2} - \frac{2}{t}$
Answer
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 $t^2$.
$$9.2\left(\frac{t^3}{3} + C\right) - \int \frac{2}{t} \, dt$$
Step 6
Factor out the constant $-2$ from the second integral.
$$9.2\left(\frac{t^3}{3} + C\right) - 2\int \frac{1}{t} \, dt$$
Step 7
Isolate the constant $2$ from the integral.
$$9.2\left(\frac{t^3}{3} + C\right) - 2\left(\int \frac{1}{t} \, dt\right)$$
Step 8
Proceed to simplify the expression.
Step 8.1
Combine the terms $\frac{1}{3}$ and $t^3$.
$$9.2\left(\frac{t^3}{3} + C\right) - 2\left(\int \frac{1}{t} \, dt\right)$$
Step 8.2
Multiply $2$ by the integral of $\frac{1}{t}$.
$$9.2\left(\frac{t^3}{3} + C\right) - 2\int \frac{1}{t} \, dt$$
Step 9
Integrate $\frac{1}{t}$ to get the natural logarithm.
$$9.2\left(\frac{t^3}{3} + C\right) - 2\left(\ln |t| + C\right)$$
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:
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., $\int f(x) \, dx$.
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 $n \neq -1$. For example, $\int t^2 \, dt = \frac{t^3}{3} + C$.
Integral of Reciprocal: The integral of $\frac{1}{x}$ with respect to $x$ is $\ln|x| + C$, where $C$ is the constant of integration.
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 $\int k f(x) \, dx = k \int f(x) \, dx$.
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 $\ln|x|$.
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 $x \leq 0$.
