To find the antiderivatives of the function [tex]\( f(y) = -\frac{28}{y^{29}} \)[/tex], let's proceed as follows:
1. Identify the function:
[tex]\[
f(y) = -\frac{28}{y^{29}}
\][/tex]
2. Rewrite using exponents:
We can rewrite the function with a negative exponent for ease of integration:
[tex]\[
f(y) = -28 y^{-29}
\][/tex]
3. Apply the power rule for integration:
The power rule for integration states that if [tex]\( f(y) = y^n \)[/tex], then the antiderivative [tex]\( F(y) \)[/tex] is given by:
[tex]\[
\int y^n \, dy = \frac{y^{n+1}}{n+1} + C
\][/tex]
where [tex]\( C \)[/tex] is the constant of integration.
4. Integrate the function:
In our case, [tex]\( n = -29 \)[/tex]. Applying the power rule:
[tex]\[
\int -28 y^{-29} \, dy = -28 \int y^{-29} \, dy
\][/tex]
[tex]\[
= -28 \left( \frac{y^{-29+1}}{-29+1} \right) + C
\][/tex]
Simplify the exponent and the denominator:
[tex]\[
= -28 \left( \frac{y^{-28}}{-28} \right) + C
\][/tex]
[tex]\[
= -28 \left( \frac{y^{-28}}{-28} \right) + C
\][/tex]
5. Simplify the expression:
The [tex]\(-28\)[/tex] in the numerator and denominator cancels out:
[tex]\[
= y^{-28} + C
\][/tex]
Therefore, the antiderivatives of [tex]\( f(y) = -\frac{28}{y^{29}} \)[/tex] are:
[tex]\[
F(y) = y^{-28} + C
\][/tex]
where [tex]\( C \)[/tex] is the constant of integration.
