Find the Antiderivative f(y)=-18/(y^19)
You are being asked to determine the antiderivative, or the indefinite integral, of a given function f(y) with respect to the variable y. The function f(y) is provided as -18/(y^19), which is a rational function where the numerator is a constant (-18) and the denominator is y raised to the 19th power. The task is to find a function F(y) such that the derivative of F(y) with respect to y equals f(y). This involves applying the rules of integral calculus to find the expression for F(y).
$f \left(\right. y \left.\right) = \frac{- 18}{y^{19}}$
Answer
Solution:
Step 1:
To determine the antiderivative of the function $f(y)$, we integrate $f(y)$ with respect to $y$: $F(y) = \int f(y) \, dy$.
Step 2:
Write down the integral that needs to be solved: $F(y) = \int \frac{-18}{y^{19}} \, dy$.
Step 3:
Extract the negative sign from the integral: $\int -\frac{18}{y^{19}} \, dy$.
Step 4:
Since $-1$ does not depend on $y$, take it outside the integral: $- \int \frac{18}{y^{19}} \, dy$.
Step 5:
As $18$ is a constant, it can also be factored out of the integral: $- \left( 18 \int \frac{1}{y^{19}} \, dy \right)$.
Step 6:
Begin simplifying the integral expression.
Step 6.1:
Combine the constant $18$ with $-1$: $-18 \int \frac{1}{y^{19}} \, dy$.
Step 6.2:
Rewrite the integrand by expressing $y^{19}$ with a negative exponent: $-18 \int y^{-19} \, dy$.
Step 6.3:
Simplify the integrand using exponent rules.
Step 6.3.1:
Apply the power rule $(a^m)^n = a^{mn}$: $-18 \int y^{-19} \, dy$.
Step 6.3.2:
Calculate $19 \times -1$: $-18 \int y^{-19} \, dy$.
Step 7:
Integrate $y^{-19}$ using the Power Rule: $-18 \left( -\frac{1}{18} y^{-18} + C \right)$.
Step 8:
Simplify the resulting expression.
Step 8.1:
Combine terms.
Step 8.1.1:
Combine $y^{-18}$ with $\frac{1}{18}$: $-18 \left( -\frac{y^{-18}}{18} + C \right)$.
Step 8.1.2:
Apply the negative exponent rule $b^{-n} = \frac{1}{b^n}$: $-18 \left( -\frac{1}{18y^{18}} + C \right)$.
Step 8.2:
Simplify the expression: $-18 \left( -\frac{1}{18y^{18}} \right) + C$.
Step 8.3:
Further simplify.
Step 8.3.1:
Multiply $-1$ by $-18$: $18 \frac{1}{18y^{18}} + C$.
Step 8.3.2:
Combine $18$ with $\frac{1}{18y^{18}}$: $\frac{18}{18y^{18}} + C$.
Step 8.3.3:
Cancel out the common factor of $18$.
Step 8.3.3.1:
Cancel the $18$: $\frac{\cancel{18}}{\cancel{18}y^{18}} + C$.
Step 8.3.3.2:
Write the simplified expression: $\frac{1}{y^{18}} + C$.
Step 9:
Conclude with the antiderivative of $f(y) = \frac{-18}{y^{19}}$: $F(y) = \frac{1}{y^{18}} + C$.
Knowledge Notes:
To solve for the antiderivative (indefinite integral) of a function, we follow these steps:
Integration: The process of finding the antiderivative is known as integration. The antiderivative of a function $f(y)$ is another function $F(y)$ such that $F'(y) = f(y)$.
Constants and Integration: Constants can be factored out of the integral. If $k$ is a constant and $f(y)$ is a function of $y$, then $\int k \cdot f(y) \, dy = k \cdot \int f(y) \, dy$.
Negative Exponents: A term with a negative exponent in the integrand, such as $y^{-n}$, can be integrated using the Power Rule for integration.
Power Rule for Integration: For any real number $n \neq -1$, the integral of $y^n$ with respect to $y$ is $\int y^n \, dy = \frac{1}{n+1} y^{n+1} + C$, where $C$ is the constant of integration.
Simplifying Expressions: Simplifying expressions involves combining like terms, factoring out common factors, and applying exponent rules such as $b^{-n} = \frac{1}{b^n}$.
Constant of Integration: When finding an indefinite integral, we add a constant term $C$ because the derivative of a constant is zero, and hence the original function could have had any constant value added to it.
