Answer
a) $\int f(x)dx= -\frac{25}{3} x^{-3}+C$
b) $\frac{d (\int f(x)dx)}{dx}= \frac{25}{x^4}$
Work Step by Step
$f(x)=\frac{25}{x^4}$
$(a)\int f(x)dx= \int \frac{25}{x^4}dx$
$\int f(x)dx=
25\int x^{-4}dx$
$\int f(x)dx= 25(\frac{x^{-4+1}}{-4+1})+C$
$\int f(x)dx= 25(\frac{x^{-3}}{-3})+C$
$\int f(x)dx= -\frac{25}{3} x^{-3}+C$
(b) $\frac{d (\int f(x)dx)}{dx}$
= $\frac{d( -\frac{25}{3}x^{-3}+C)}{dx}$
= $\frac{d(-\frac{25}{3}x^{-3})}{dx}+\frac{d(C)}{dx}$
=$ -\frac{25}{3} \frac{d( x^{-3})}{dx}+0$
=$-\frac{25}{3}(-3x^{-4})$
= $25x^{-4}=\frac{25}{x^{4}}$
