Find the Antiderivative f(x)=9x^9-2x^6+10x^3

Solution:

Step 1

Identify the antiderivative $F(x)$ by integrating the given function $f(x)$.

$$F(x)=\int f(x)dx$$

Step 2

Write down the integral that needs to be solved.

$$F(x)=\int (9x^9-2x^6+10x^3)dx$$

Step 3

Decompose the integral into separate integrals for each term.

$$\int 9x^{9}dx-\int 2x^{6}dx+\int 10x^{3}dx$$

Step 4

Extract the constant coefficient from the integral of $x^9$.

$$9\int x^{9}dx-\int 2x^{6}dx+\int 10x^{3}dx$$

Step 5

Apply the Power Rule to integrate $x^9$.

$$9(\frac{x^{10}}{10}+C)-\int 2x^{6}dx+\int 10x^{3}dx$$

Step 6

Extract the constant coefficient from the integral of $x^6$.

$$9(\frac{x^{10}}{10}+C)-2\int x^{6}dx+\int 10x^{3}dx$$

Step 7

Apply the Power Rule to integrate $x^6$.

$$9(\frac{x^{10}}{10}+C)-2(\frac{x^7}{7}+C)+\int 10x^{3}dx$$

Step 8

Extract the constant coefficient from the integral of $x^3$.

$$9(\frac{x^{10}}{10}+C)-2(\frac{x^7}{7}+C)+10\int x^{3}dx$$

Step 9

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

$$9(\frac{x^{10}}{10}+C)-2(\frac{x^7}{7}+C)+10(\frac{x^4}{4}+C)$$

Step 10

Combine and simplify the terms.

Step 10.1

Combine the terms with their coefficients.

$$\frac{9x^{10}}{10}-\frac{2x^7}{7}+\frac{10x^4}{4}+C$$

Step 10.2

Simplify the expression by reducing fractions where possible.

Step 10.2.1

Simplify the coefficient of $x^4$.

$$\frac{9x^{10}}{10}-\frac{2x^7}{7}+\frac{5x^4}{2}+C$$

Step 11

Arrange the terms in descending order of power.

$$\frac{9}{10}{x}^{10}-\frac{2}{7}{x}^7+\frac{5}{2}{x}^4+C$$

Step 12

Present the final antiderivative of the function $f(x)=9x^9-2x^6+10x^3$.

$$F(x)=\frac{9}{10}{x}^{10}-\frac{2}{7}{x}^7+\frac{5}{2}{x}^4+C$$

Knowledge Notes:

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

1. Indefinite Integral: The antiderivative of a function $f(x)$ is represented by the indefinite integral $\int f(x)dx$.

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. For example, $\int k\cdot f(x)dx=k\cdot \int f(x)dx$ where $k$ is a constant.

3. Power Rule for Integration: The Power Rule states that the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1}+C$, for any real number $n\ne -1$.

4. Linearity of Integration: The integral of a sum or difference of functions is the sum or difference of their integrals. For example, $\int [f(x)\pm g(x)]dx=\int f(x)dx\pm \int g(x)dx$.

5. Arbitrary Constant of Integration: 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 applying the Power Rule, it is often necessary to simplify the expression by combining like terms, reducing fractions, and arranging the terms in a standard form, usually in descending order of the power of $x$.