Find the Antiderivative f(x)=9x^9-2x^6+10x^3
The problem asks for the calculation of the antiderivative (also known as the indefinite integral) of the polynomial function f(x) = 9x^9 - 2x^6 + 10x^3. An antiderivative is a function whose derivative is the original function. In this case, you are expected to determine a function F(x) such that the derivative F'(x) equals the given polynomial function f(x). The question requires knowledge of basic integral calculus, specifically the rules for integrating powers of x.
$f \left(\right. x \left.\right) = 9 x^{9} - 2 x^{6} + 10 x^{3}$
Answer
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 $x^9$.
\[ 9\left( \frac{x^{10}}{10} + C \right) - \int 2x^6 \, dx + \int 10x^3 \, dx \]
Step 6
Extract the constant coefficient from the integral of $x^6$.
\[ 9\left( \frac{x^{10}}{10} + C \right) - 2\int x^6 \, dx + \int 10x^3 \, dx \]
Step 7
Apply the Power Rule to integrate $x^6$.
\[ 9\left( \frac{x^{10}}{10} + C \right) - 2\left( \frac{x^7}{7} + C \right) + \int 10x^3 \, dx \]
Step 8
Extract the constant coefficient from the integral of $x^3$.
\[ 9\left( \frac{x^{10}}{10} + C \right) - 2\left( \frac{x^7}{7} + C \right) + 10\int x^3 \, dx \]
Step 9
Apply the Power Rule to integrate $x^3$.
\[ 9\left( \frac{x^{10}}{10} + C \right) - 2\left( \frac{x^7}{7} + C \right) + 10\left( \frac{x^4}{4} + C \right) \]
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:
Indefinite Integral: The antiderivative of a function $f(x)$ is represented by the indefinite integral $\int f(x) \, dx$.
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.
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 $n \neq -1$.
Linearity of Integration: The integral of a sum or difference of functions is the sum or difference of their integrals. For example, $\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx$.
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.
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 $x$.
