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, $\int t^7 \, dt$.

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

Step 6:

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

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

Step 7:

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

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

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 $n \neq -1$, 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.