Let's find the antiderivatives of the function [tex]\( f(y) = -\frac{24}{y^{25}} \)[/tex].
### Step 1: Rewrite the Function
Rewrite the function in a more convenient form:
[tex]\[ f(y) = -24y^{-25} \][/tex]
### Step 2: Integrate
To find the antiderivative, we need to perform the integration:
[tex]\[ \int -24y^{-25} \, dy \][/tex]
We use the power rule for integration, which states:
[tex]\[ \int y^n \, dy = \frac{y^{n+1}}{n+1} + C \][/tex]
for any [tex]\( n \neq -1 \)[/tex]. Here, [tex]\( n = -25 \)[/tex]:
[tex]\[ \int -24y^{-25} \, dy = -24 \int y^{-25} \, dy \][/tex]
[tex]\[ \int y^{-25} \, dy = \frac{y^{-25+1}}{-25+1} = \frac{y^{-24}}{-24} \][/tex]
Therefore:
[tex]\[ -24 \int y^{-25} \, dy = -24 \cdot \frac{y^{-24}}{-24} = y^{-24} \][/tex]
Don't forget to add the constant of integration [tex]\( C \)[/tex]:
[tex]\[ \int -24y^{-25} \, dy = y^{-24} + C \][/tex]
So, the antiderivative [tex]\( F(y) \)[/tex] is:
[tex]\[ F(y) = y^{-24} + C \][/tex]
### Step 3: Check the Work
To verify, let's take the derivative of [tex]\( F(y) \)[/tex]:
[tex]\[ \frac{d}{dy}\left( y^{-24} + C \right) \][/tex]
Since the derivative of a constant [tex]\( C \)[/tex] is zero:
[tex]\[ \frac{d}{dy}\left( y^{-24} \right) = -24y^{-25} \][/tex]
So:
[tex]\[ \frac{d}{dy}\left( y^{-24} + C \right) = -24y^{-25} \][/tex]
Which matches the original function [tex]\( f(y) = -24y^{-25} \)[/tex].
### Conclusion
The antiderivatives of [tex]\( f(y) = -\frac{24}{y^{25}} \)[/tex] are:
[tex]\[ F(y) = y^{-24} + C \][/tex]
This confirms our solution is correct.
