Find the Antiderivative f(t)=2t^2+3t^3+4t^4

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 $t^2$.

$$2(\frac{t^3}{3}+C)+\int 3t^{3}dt+\int 4t^{4}dt$$

Step 6:

Extract the constant coefficient 3 from the second integral.

$$2(\frac{t^3}{3}+C)+3\int t^{3}dt+\int 4t^{4}dt$$

Step 7:

Apply the Power Rule to integrate $t^3$.

$$2(\frac{t^3}{3}+C)+3(\frac{t^4}{4}+C)+\int 4t^{4}dt$$

Step 8:

Extract the constant coefficient 4 from the third integral.

$$2(\frac{t^3}{3}+C)+3(\frac{t^4}{4}+C)+4\int t^{4}dt$$

Step 9:

Apply the Power Rule to integrate $t^4$.

$$2(\frac{t^3}{3}+C)+3(\frac{t^4}{4}+C)+4(\frac{t^5}{5}+C)$$

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:

1. 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.

2. Constant Multiple Rule: When a constant multiplies a function, it can be factored out of the integral. For example, $\int cf(t)dt=c\int f(t)dt$.

3. 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 $n\ne -1$.

4. Integration of Sum: The integral of a sum of functions is equal to the sum of their integrals. For example, $\int (f(t)+g(t))dt=\int f(t)dt+\int g(t)dt$.

5. 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.

6. Simplification: After integrating, it's often necessary to simplify the expression by combining like terms and constants to achieve the final antiderivative.