Find the Antiderivative f(t)=2t^2+3t^3+4t^4
The given problem is asking for the computation of the antiderivative, also known as an indefinite integral, of the polynomial function f(t) = 2t^2 + 3t^3 + 4t^4. The antiderivative is a function whose derivative is the original function f(t). In other words, it requires finding a function F(t) such that F'(t) = f(t). This involves applying integral calculus techniques to determine the formula for the function whose derivative yields the given polynomial. The challenge here is to perform the correct integration operation for each term of the polynomial individually, and combine those results into a general expression for the antiderivative.
$f \left(\right. t \left.\right) = 2 t^{2} + 3 t^{3} + 4 t^{4}$
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 the integral of the given function.
$$F(t) = \int (2t^2 + 3t^3 + 4t^4) \, dt$$
Step 3:
Decompose the integral into separate terms.
$$\int 2t^2 \, dt + \int 3t^3 \, dt + \int 4t^4 \, dt$$
Step 4:
Extract the constant coefficient 2 from the first integral.
$$2\int t^2 \, dt + \int 3t^3 \, dt + \int 4t^4 \, dt$$
Step 5:
Apply the Power Rule to integrate $t^2$.
$$2\left(\frac{t^3}{3} + C\right) + \int 3t^3 \, dt + \int 4t^4 \, dt$$
Step 6:
Extract the constant coefficient 3 from the second integral.
$$2\left(\frac{t^3}{3} + C\right) + 3\int t^3 \, dt + \int 4t^4 \, dt$$
Step 7:
Apply the Power Rule to integrate $t^3$.
$$2\left(\frac{t^3}{3} + C\right) + 3\left(\frac{t^4}{4} + C\right) + \int 4t^4 \, dt$$
Step 8:
Extract the constant coefficient 4 from the third integral.
$$2\left(\frac{t^3}{3} + C\right) + 3\left(\frac{t^4}{4} + C\right) + 4\int t^4 \, dt$$
Step 9:
Apply the Power Rule to integrate $t^4$.
$$2\left(\frac{t^3}{3} + C\right) + 3\left(\frac{t^4}{4} + C\right) + 4\left(\frac{t^5}{5} + C\right)$$
Step 10:
Proceed to simplify the expression.
Step 10.1:
Combine terms and constants.
$$\frac{2t^3}{3} + \frac{3t^4}{4} + 4\left(\frac{t^5}{5}\right) + C$$
Step 10.2:
Further simplify the expression.
Step 10.2.1:
Merge the coefficient $\frac{1}{5}$ with $t^5$.
$$\frac{2t^3}{3} + \frac{3t^4}{4} + \frac{4t^5}{5} + C$$
Step 10.2.2:
Combine the constant 4 with $\frac{t^5}{5}$.
$$\frac{2t^3}{3} + \frac{3t^4}{4} + \frac{4t^5}{5} + C$$
Step 10.3:
Reorganize the terms in the expression.
$$\frac{2}{3}t^3 + \frac{3}{4}t^4 + \frac{4}{5}t^5 + C$$
Step 11:
Conclude with the antiderivative of the function $f(t) = 2t^2 + 3t^3 + 4t^4$.
$$F(t) = \frac{2}{3}t^3 + \frac{3}{4}t^4 + \frac{4}{5}t^5 + C$$
Knowledge Notes:
The process of finding the antiderivative, also known as the indefinite integral, involves integrating a given function. Here are the relevant knowledge points:
Indefinite Integral: The antiderivative of a function $f(t)$ is represented by $\int f(t) \, dt$, which is the most general form of the integral without specific limits.
Constant Multiple Rule: When a constant multiplies a function, it can be factored out of the integral. For example, $\int cf(t) \, dt = c \int f(t) \, dt$.
Power Rule for Integration: To integrate a power of $t$, use the formula $\int t^n \, dt = \frac{t^{n+1}}{n+1} + C$, where $n$ is a real number and $n \neq -1$.
Integration of Sum: The integral of a sum of functions is equal to the sum of their integrals. For example, $\int (f(t) + g(t)) \, dt = \int f(t) \, dt + \int g(t) \, dt$.
Arbitrary Constant: When finding the indefinite integral, an arbitrary constant $C$ is added to the result because the derivative of a constant is zero, and thus the original function could have had any constant added to it.
Simplification: After integrating, it's often necessary to simplify the expression by combining like terms and constants to achieve the final antiderivative.
