Find the Antiderivative f(t)=3t^3+17
The question asks you to determine the antiderivative (also known as the indefinite integral) of the function f(t) = 3t^3 + 17. The antiderivative involves finding a function whose derivative is the given function f(t). Essentially, it is asking for the integral with respect to t, which would result in a new function F(t) that satisfies the condition that dF(t)/dt = f(t), plus a constant of integration, since taking the derivative of a constant yields zero.
$f \left(\right. t \left.\right) = 3 t^{3} + 17$
Answer
Solution:
Step 1:
Identify the antiderivative $F(t)$ by integrating the given function $f(t)$.
$$ F(t) = \int f(t) \, dt $$
Step 2:
Write down the integral that needs to be solved.
$$ F(t) = \int (3t^3 + 17) \, dt $$
Step 3:
Decompose the integral into simpler integrals.
$$ \int 3t^3 \, dt + \int 17 \, dt $$
Step 4:
Extract the constant $3$ from the first integral.
$$ 3\int t^3 \, dt + \int 17 \, dt $$
Step 5:
Apply the power rule for integration to $t^3$.
$$ 3\left( \frac{t^4}{4} + C \right) + \int 17 \, dt $$
Step 6:
Integrate the constant $17$ with respect to $t$.
$$ 3\left( \frac{t^4}{4} + C \right) + 17t + C $$
Step 7:
Simplify the expression.
Step 7.1:
Combine the constant $\frac{1}{4}$ with $t^4$.
$$ 3\left( \frac{t^4}{4} + C \right) + 17t + C $$
Step 7.2:
Simplify the expression further.
$$ \frac{3t^4}{4} + 17t + C $$
Step 7.3:
Reorder the terms if necessary.
$$ \frac{3t^4}{4} + 17t + C $$
Step 8:
Conclude with the antiderivative of the function $f(t) = 3t^3 + 17$.
$$ F(t) = \frac{3t^4}{4} + 17t + C $$
Knowledge Notes:
To solve for the antiderivative of a polynomial function, one can use the following knowledge points:
Indefinite Integral: The antiderivative, or indefinite integral, of a function $f(t)$ is denoted by $\int f(t) \, dt$ and represents the family of all functions whose derivative is $f(t)$.
Linearity of Integration: The integral of a sum of functions is equal to the sum of their integrals. This property allows us to split a complex integral into simpler parts.
Constant Multiple Rule: A constant multiple outside the integral can be factored out, which simplifies the integration process.
Power Rule for Integration: For any real number $n \neq -1$, the integral of $t^n$ with respect to $t$ is $\frac{t^{n+1}}{n+1} + C$, where $C$ is the constant of integration.
Integration of a Constant: The integral of a constant $a$ with respect to $t$ is $at + C$, where $C$ is the constant of integration.
Simplification: After integrating, it's important to simplify the expression by combining like terms and constants.
Constant of Integration: When finding the indefinite integral, we must add a constant of integration $C$ because the derivative of a constant is zero, and thus the original function could have any constant added to it.
