Find the Antiderivative v(t)=4t+5
This question asks you to determine the antiderivative (or indefinite integral) of the given function v(t) = 4t + 5. The antiderivative represents the function whose derivative would yield v(t). You are expected to find a function V(t), such that when differentiated with respect to t, it equals the velocity function v(t) = 4t + 5.
$v \left(\right. t \left.\right) = 4 t + 5$
Answer
Solution:
Step 1:
Identify the antiderivative $V(t)$ by integrating the velocity function $v(t)$.
$$V(t) = \int v(t) \, dt$$
Step 2:
Write down the integral that needs to be solved.
$$V(t) = \int (4t + 5) \, dt$$
Step 3:
Decompose the integral into simpler parts.
$$\int 4t \, dt + \int 5 \, dt$$
Step 4:
Extract the constant factor $4$ from the first integral.
$$4 \int t \, dt + \int 5 \, dt$$
Step 5:
Apply the power rule to integrate $t$ with respect to $t$.
$$4 \left( \frac{t^2}{2} + C \right) + \int 5 \, dt$$
Step 6:
Integrate the constant $5$ with respect to $t$.
$$4 \left( \frac{t^2}{2} + C \right) + 5t + C$$
Step 7:
Simplify the expression.
Step 7.1:
Combine the terms involving $t^2$.
$$4 \left( \frac{t^2}{2} + C \right) + 5t + C$$
Step 7.2:
Final simplification.
$$2t^2 + 5t + C$$
Step 8:
Present the antiderivative of $v(t) = 4t + 5$.
$$V(t) = 2t^2 + 5t + C$$
Knowledge Notes:
To solve for the antiderivative of a function, one must integrate the function with respect to its variable. The steps involved in this process typically include:
Indefinite Integral: The antiderivative is found by integrating the function without specifying the limits of integration. This is represented by the integral symbol $\int$ followed by the function and the differential of the variable, e.g., $\int f(x) \, dx$.
Linear Property of Integration: This property allows us to split the integral of a sum into the sum of integrals and to factor out constants. For example, $\int (af(x) + bg(x)) \, dx = a\int f(x) \, dx + b\int g(x) \, dx$.
Power Rule for Integration: When integrating a power of $x$, the antiderivative is found by increasing the exponent by one and then dividing by the new exponent, as long as the exponent is not $-1$. The formula is $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$.
Constant Rule for Integration: The integral of a constant $a$ with respect to $x$ is $ax + C$, where $C$ is the constant of integration.
Simplification: After applying the rules of integration, the expression is often simplified by combining like terms and factoring, if possible.
Constant of Integration: Since the derivative of a constant is zero, when finding the antiderivative, an arbitrary constant $C$ is added to represent any possible constant term that was lost during differentiation.
