Find the Antiderivative y=2x
The problem provided is asking for the calculation of the antiderivative (also known as the indefinite integral) of the function y = 2x. This involves finding a function whose derivative is 2x. Essentially, you are being asked to reverse the differentiation process and identify a function F(x), such that when it is differentiated, it results in the given function, which in this case is 2x. The solution will involve the basic rules of integration and will include a constant of integration since the antiderivative is not unique—any constant added to a function's antiderivative is also an antiderivative of that function.
$y = 2 x$
Answer
Solution:
Step 1:
Express the given function $y = 2x$ in function notation: $f(x) = 2x$.
Step 2:
Identify the antiderivative $F(x)$ by integrating the function $f(x)$: $F(x) = \int f(x) \, dx$.
Step 3:
Prepare to solve the integral: $F(x) = \int 2x \, dx$.
Step 4:
Extract the constant $2$ from the integral: $2 \int x \, dx$.
Step 5:
Apply the Power Rule for integration to $x$: $2 \left( \frac{x^2}{2} + C \right)$.
Step 6:
Simplify the expression.
Step 6.1:
Rewrite the expression as $2 \left( \frac{1}{2} \right) x^2 + 2C$.
Step 6.2:
Perform the simplification.
Step 6.2.1:
Combine the constants: $\frac{2}{2} x^2 + 2C$.
Step 6.2.2:
Eliminate the common factor.
Step 6.2.2.1:
Cancel out the common factors: $\frac{\cancel{2}}{\cancel{2}} x^2 + 2C$.
Step 6.2.2.2:
Present the simplified expression: $x^2 + 2C$.
Step 6.2.3:
Recognize that $1 \cdot x^2 = x^2$: $x^2 + 2C$.
Step 7:
Conclude with the antiderivative of $f(x) = 2x$: $F(x) = x^2 + C$.
Knowledge Notes:
Antiderivative: An antiderivative of a function $f(x)$ is a function $F(x)$ whose derivative is $f(x)$. It is often represented by the integral symbol $\int$ followed by the function and the differential $dx$.
Function Notation: Functions are often written in the form $f(x)$, where $f$ is the function name and $x$ is the variable.
Indefinite Integral: The indefinite integral, represented by $\int f(x) \, dx$, is the general form of the antiderivative and includes a constant of integration $C$.
Power Rule for Integration: The Power Rule states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, where $n$ is a real number and $n \neq -1$.
Constant Multiple Rule: A constant multiple outside of an integral can be factored out, making the integration process simpler. For example, $\int k \cdot f(x) \, dx = k \cdot \int f(x) \, dx$, where $k$ is a constant.
Simplification: Combining like terms and canceling common factors are standard algebraic techniques used to simplify expressions.
Constant of Integration: When finding an indefinite integral, the constant of integration $C$ is added to represent the family of all antiderivatives. It accounts for the fact that there are infinitely many antiderivatives, each differing by a constant.
