Find the Antiderivative f(t)=7t^7+8t^8+9t^9

Solution:

Step 1:

Identify the function $f(t)$ whose antiderivative is to be found. The antiderivative, denoted as $F(t)$, is the integral of $f(t)$ with respect to $t$.

$$F(t)=\int f(t)dt$$

Step 2:

Write down the integral that needs to be solved.

$$F(t)=\int (7t^7+8t^8+9t^9)dt$$

Step 3:

Decompose the integral into a sum of integrals for each term.

$$\int 7t^{7}dt+\int 8t^{8}dt+\int 9t^{9}dt$$

Step 4:

Extract the constant coefficients from each integral.

$$7\int t^{7}dt+8\int t^{8}dt+9\int t^{9}dt$$

Step 5:

Apply the Power Rule for integration to the first term, $\int t^{7}dt$.

$$7(\frac{t^8}{8}+C)+8\int t^{8}dt+9\int t^{9}dt$$

Step 6:

Apply the Power Rule to the second term, $\int t^{8}dt$.

$$7(\frac{t^8}{8}+C)+8(\frac{t^9}{9}+C)+9\int t^{9}dt$$

Step 7:

Apply the Power Rule to the third term, $\int t^{9}dt$.

$$7(\frac{t^8}{8}+C)+8(\frac{t^9}{9}+C)+9(\frac{t^{10}}{10}+C)$$

Step 8:

Combine the terms and simplify the expression.

$$\frac{7t^8}{8}+\frac{8t^9}{9}+\frac{9t^{10}}{10}+C$$

Step 9:

Reorder the terms to present the final antiderivative.

$$F(t)=\frac{7}{8}{t}^8+\frac{8}{9}{t}^9+\frac{9}{10}{t}^{10}+C$$

Step 10:

Conclude with the antiderivative of the given function $f(t)=7t^7+8t^8+9t^9$.

$$F(t)=\frac{7}{8}{t}^8+\frac{8}{9}{t}^9+\frac{9}{10}{t}^{10}+C$$

Knowledge Notes:

The process of finding the antiderivative involves several key knowledge points:

1. Indefinite Integral: The antiderivative of a function $f(t)$ is also known as the indefinite integral, represented by $\int f(t)dt$. It is the reverse process of differentiation.

2. Constant Multiple Rule: When a constant is multiplied by a function, the integral of the product is the constant multiplied by the integral of the function. Mathematically, $\int cf(t)dt=c\int f(t)dt$ where $c$ is a constant.

3. Power Rule for Integration: This rule is used to integrate powers of $t$. The rule states that $\int t^{n}dt=\frac{t^{n+1}}{n+1}+C$ for any real number $n\ne -1$, where $C$ is the constant of integration.

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

5. Constant of Integration: When finding the indefinite integral, a constant $C$ is added to represent the family of all antiderivatives, since the derivative of a constant is zero.

These principles are applied in the solution to find the antiderivative of the given polynomial function.