Find the Antiderivative f(y)=-9/(y^10)
The question asks you to determine the antiderivative, which is the function whose derivative is equal to the given function f(y)=-9/(y^10). The process of finding this function involves integrating the given function with respect to the variable 'y'. You are looking for a function F(y) such that when differentiated with respect to 'y', it yields the original function f(y)=-9/(y^10). This process is also known as finding the indefinite integral of the given function.
$f \left(\right. y \left.\right) = \frac{- 9}{y^{10}}$
Answer
Solution:
Step 1:
Identify the antiderivative $F(y)$ by integrating the given function $f(y)$.
$$F(y) = \int f(y) \, dy$$
Step 2:
Write down the integral that needs to be solved.
$$F(y) = \int \frac{-9}{y^{10}} \, dy$$
Step 3:
Extract the negative sign from the integral.
$$\int -\frac{9}{y^{10}} \, dy$$
Step 4:
Since $-1$ is a constant, take it outside the integral.
$$- \int \frac{9}{y^{10}} \, dy$$
Step 5:
Also, take the constant $9$ outside the integral.
$$-9 \int \frac{1}{y^{10}} \, dy$$
Step 6:
Begin simplifying the integral.
Step 6.1:
Combine the constants.
$$-9 \int \frac{1}{y^{10}} \, dy$$
Step 6.2:
Rewrite the integrand using negative exponents.
$$-9 \int y^{-10} \, dy$$
Step 6.3:
Prepare the integrand for applying the power rule.
Step 6.3.1:
Recall the rule for exponents: $(a^m)^n = a^{mn}$.
$$-9 \int y^{10 \cdot (-1)} \, dy$$
Step 6.3.2:
Perform the multiplication of exponents.
$$-9 \int y^{-10} \, dy$$
Step 7:
Apply the power rule for integration, $\int y^n \, dy = \frac{y^{n+1}}{n+1} + C$ for $n \neq -1$.
$$-9 \left( -\frac{1}{9} y^{-9} + C \right)$$
Step 8:
Simplify the expression.
Step 8.1:
Start simplifying the terms.
Step 8.1.1:
Combine the constants and the variable term.
$$-9 \left( -\frac{y^{-9}}{9} + C \right)$$
Step 8.1.2:
Apply the negative exponent rule: $b^{-n} = \frac{1}{b^n}$.
$$-9 \left( -\frac{1}{9y^9} + C \right)$$
Step 8.2:
Simplify the negative signs and constants.
$$-9 \left( -\frac{1}{9y^9} \right) + C$$
Step 8.3:
Final simplification steps.
Step 8.3.1:
Multiply the negative signs.
$$9 \frac{1}{9y^9} + C$$
Step 8.3.2:
Simplify the fraction by canceling out common factors.
$$\frac{9}{9y^9} + C$$
Step 8.3.3:
Cancel the 9s.
$$\frac{1}{y^9} + C$$
Step 9:
Conclude with the antiderivative of $f(y) = \frac{-9}{y^{10}}$.
$$F(y) = \frac{1}{y^9} + C$$
Knowledge Notes:
Antiderivative: The antiderivative of a function $f(y)$ is another function $F(y)$ such that $F'(y) = f(y)$. It is found by the process of integration.
Indefinite Integral: The indefinite integral of a function $f(y)$ with respect to $y$ is denoted as $\int f(y) \, dy$ and represents the collection of all antiderivatives of $f(y)$.
Constants in Integration: Constants can be moved outside the integral sign because they do not depend on the variable of integration.
Negative Exponents: The expression $y^{-n}$ is equivalent to $\frac{1}{y^n}$.
Power Rule for Integration: For any real number $n \neq -1$, the integral of $y^n$ with respect to $y$ is $\frac{y^{n+1}}{n+1} + C$, where $C$ is the constant of integration.
Simplification: In the process of integration, simplifying the expression involves combining like terms, canceling out common factors, and applying exponent rules to arrive at the simplest form of the antiderivative.
