Find the Antiderivative f(y)=-19/(y^20)
The question is asking for the calculation of the antiderivative, also known as the indefinite integral, of a given function with respect to a variable 'y'. Specifically, the function provided is f(y) = -19/(y^20), which is a rational function where the variable 'y' is in the denominator raised to the 20th power, and the constant -19 is in the numerator. The task is to find a function F(y) such that its derivative with respect to 'y' gives back the original function f(y).
$f \left(\right. y \left.\right) = \frac{- 19}{y^{20}}$
Answer
Solution:
Step 1:
Identify the antiderivative $F(y)$ by integrating the function $f(y)$.
$$ F(y) = \int f(y) \, dy $$
Step 2:
Write the integral that needs to be solved.
$$ F(y) = \int \frac{-19}{y^{20}} \, dy $$
Step 3:
Extract the negative sign from the integral.
$$ \int -\frac{19}{y^{20}} \, dy $$
Step 4:
Since $-1$ is a constant with respect to $y$, factor it out from the integral.
$$ -\int \frac{19}{y^{20}} \, dy $$
Step 5:
As $19$ is also a constant, factor it out as well.
$$ -19 \int \frac{1}{y^{20}} \, dy $$
Step 6:
Begin simplifying the integral.
Step 6.1:
Combine the constants.
$$ -19 \int \frac{1}{y^{20}} \, dy $$
Step 6.2:
Rewrite the integrand using a negative exponent.
$$ -19 \int y^{-20} \, dy $$
Step 6.3:
Apply the power rule for exponents.
Step 6.3.1:
Use the rule $(a^m)^n = a^{mn}$.
$$ -19 \int y^{20 \cdot (-1)} \, dy $$
Step 6.3.2:
Calculate $20 \cdot (-1)$.
$$ -19 \int y^{-20} \, dy $$
Step 7:
Integrate using the Power Rule for integration.
$$ -19 \left( -\frac{1}{19} y^{-19} + C \right) $$
Step 8:
Simplify the expression.
Step 8.1:
Combine terms.
$$ -19 \left( -\frac{y^{-19}}{19} + C \right) $$
Step 8.2:
Simplify the constant multiplication.
$$ -19 \left( -\frac{1}{19 y^{19}} \right) + C $$
Step 8.3:
Final simplifications.
Step 8.3.1:
Multiply through by $-1$.
$$ 19 \frac{1}{19 y^{19}} + C $$
Step 8.3.2:
Simplify the fraction.
$$ \frac{19}{19 y^{19}} + C $$
Step 8.3.3:
Cancel out common factors.
Step 8.3.3.1:
Perform the cancellation.
$$ \frac{1}{y^{19}} + C $$
Step 9:
Conclude with the antiderivative of $f(y) = \frac{-19}{y^{20}}$.
$$ F(y) = \frac{1}{y^{19}} + C $$
Knowledge Notes:
To solve for an antiderivative, or indefinite integral, of a function, we follow a series of steps that involve algebraic manipulation and the application of integration rules. In this case, we are dealing with the integral of a power function. The key knowledge points involved in this problem include:
Indefinite Integral: The antiderivative of a function, which includes an arbitrary constant $C$ because the derivative of a constant is zero.
Constant Factor Rule: Constants can be factored out of an integral, as they do not depend on the variable of integration.
Power Rule for Integration: For any real number $n \neq -1$, the integral of $x^n$ with respect to $x$ is $\frac{x^{n+1}}{n+1} + C$.
Negative Exponents: A term with a negative exponent can be rewritten as the reciprocal of the term with a positive exponent, i.e., $x^{-n} = \frac{1}{x^n}$.
Simplification: Algebraic simplification, including canceling common factors, is often necessary to arrive at the simplest form of the antiderivative.
In this problem, we applied these principles to find the antiderivative of a rational function with a negative exponent. After setting up the integral, we factored out constants and applied the power rule for integration. The final step involved algebraic simplification, including canceling common factors, to express the antiderivative in its simplest form.
