Find the Antiderivative f(y)=-10/(y^11)
The problem is asking for the computation of the antiderivative, also known as the indefinite integral, of the given function f(y) = -10/(y^11). The antiderivative is a function whose derivative is the original function provided. You are to determine the function F(y) such that F'(y) = f(y), which involves integrating the given function with respect to the variable y.
$f \left(\right. y \left.\right) = \frac{- 10}{y^{11}}$
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.
$$ F(y) = \int \frac{-10}{y^{11}} \, dy $$
Step 3:
Extract the negative sign from the integral.
$$ \int -\frac{10}{y^{11}} \, dy $$
Step 4:
Since $-1$ is a constant with respect to $y$, it can be taken out of the integral.
$$ -\int \frac{10}{y^{11}} \, dy $$
Step 5:
As $10$ is also a constant, it can be factored out as well.
$$ -10 \int \frac{1}{y^{11}} \, dy $$
Step 6:
Begin simplifying the integral expression.
Step 6.1:
Combine the constants $-1$ and $10$.
$$ -10 \int \frac{1}{y^{11}} \, dy $$
Step 6.2:
Rewrite the integrand by expressing $y^{11}$ with a negative exponent.
$$ -10 \int y^{-11} \, dy $$
Step 6.3:
Apply the exponent multiplication rule.
Step 6.3.1:
Use the rule $(a^m)^n = a^{mn}$ to simplify the exponent.
$$ -10 \int y^{-11} \, dy $$
Step 6.3.2:
Calculate the product of the exponents $11$ and $-1$.
$$ -10 \int y^{-11} \, dy $$
Step 7:
Utilize the Power Rule for integration to find the integral of $y^{-11}$.
$$ -10 \left( -\frac{1}{10} y^{-10} + C \right) $$
Step 8:
Proceed to simplify the expression.
Step 8.1:
Start simplifying the terms inside the parentheses.
Step 8.1.1:
Combine the terms $y^{-10}$ and $\frac{1}{10}$.
$$ -10 \left( -\frac{y^{-10}}{10} + C \right) $$
Step 8.1.2:
Apply the negative exponent rule to move $y^{-10}$ to the denominator.
$$ -10 \left( -\frac{1}{10y^{10}} + C \right) $$
Step 8.2:
Continue simplifying by distributing the $-10$.
$$ -10 \left( -\frac{1}{10y^{10}} \right) + C $$
Step 8.3:
Finalize the simplification process.
Step 8.3.1:
Multiply $-1$ by $-10$.
$$ 10 \frac{1}{10y^{10}} + C $$
Step 8.3.2:
Combine the constants $10$ and $\frac{1}{10y^{10}}$.
$$ \frac{10}{10y^{10}} + C $$
Step 8.3.3:
Eliminate the common factor of $10$.
Step 8.3.3.1:
Cancel out the common factor.
$$ \frac{\cancel{10}}{\cancel{10}y^{10}} + C $$
Step 8.3.3.2:
Rewrite the simplified expression.
$$ \frac{1}{y^{10}} + C $$
Step 9:
Conclude with the antiderivative of the function $f(y) = \frac{-10}{y^{11}}$.
$$ F(y) = \frac{1}{y^{10}} + C $$
Knowledge Notes:
To solve for the antiderivative of a function, we follow these steps:
Integration: The process of finding the antiderivative is called integration. The antiderivative $F(y)$ of a function $f(y)$ is found by integrating $f(y)$ with respect to $y$.
Constants: Constants can be factored out of the integral, simplifying the integration process.
Negative Exponents: Expressing terms with negative exponents can help in applying integration rules.
Power Rule for Integration: The Power Rule states that the integral of $y^n$ with respect to $y$ is $\frac{y^{n+1}}{n+1}$, provided $n \neq -1$.
Simplification: After applying the Power Rule, it's important to simplify the expression, including canceling out common factors.
Integration Constant: When finding the indefinite integral, don't forget to add the constant of integration $C$, which represents the family of all antiderivatives.
