# Consider the following. f(x)= \frac{25}{x^4} (a) Write the formula for \int f(x)\ dx. (Use C for...

## Question:

Consider the following.

{eq}f(x)= \frac{25}{x^4} {/eq}

(a) Write the formula for {eq}\int f(x)\ dx. {/eq} (Use C for the constant of integration.)

(b) Write the formula for {eq}\frac{d}{dx} \int f(x)\ dx. {/eq}

## Calculus Theorem:

In 2D calculus, if there is a function with one independent variable {eq}F(x) {/eq} and its first derivative, {eq}f'(x) {/eq} the following theorem is true:

{eq}\displaystyle \int \ f'(x) \;dx= \ F(x) \; + \; C \\ {/eq}

## Answer and Explanation:

(a) Write the formula for {eq}\int f(x)\ dx. {/eq} (Use C for the constant of integration.)

The function is:

{eq}\displaystyle \ f(x) = \frac{25}{x^4} \; \Rightarrow \; \ f(x) = 25x^{-4} \\ {/eq}

The integral is:

{eq}\displaystyle \int \; 25x^{-4} \;dx= \frac{ 25x^{-4+1}}{ -4+1 } + \; C \\ \displaystyle \int \; 25x^{-4} \;dx= \frac{ 25x^{-3}}{ -3 } + \; C \\ \displaystyle \int \; 25x^{-4} \;dx= -\frac{ 25}{ 3x^{3} } + \; C \\ {/eq}

(b) Write the formula for {eq}\displaystyle \frac{d}{dx} \int f(x)\ dx= \ f(x) \\ \displaystyle \frac{d}{dx} \int \; \frac{25}{x^4} \; \ dx= \frac{25}{x^4}\\ {/eq}