# Question Video: Finding the Expression of a Function given the Expression of Its Slope and a Point on the Function Using Indefinite Integration Mathematics

The slope at the point (π₯, π¦) on the graph of a function is 6π^(π₯) + 2. What is π(π₯), given that π(ln 5) = 1?

02:30

### Video Transcript

The slope at the point π₯, π¦ on the graph of a function is six π to the power of π₯ plus two. What is π of π₯, given that π of the natural log of five equals one?

Weβve been given information about the slope at a general point π₯, π¦. And, actually, if we recall, we can find an expression for the slope of the graph of a function π of π₯ by finding its derivative. This means given a function π of π₯, its derivative π prime of π₯ at any point π₯, π¦ here will be defined by six π to the power of π₯ plus two. Now, what do we know about the relationship between π of π₯ and π prime of π₯?

Well, π prime of π₯ is the derivative of our function. But we also know that the opposite of differentiating is integrating. And so, if we integrate π prime of π₯ with respect to π₯, we will find a general solution for π of π₯. Weβll then use the fact that π of the natural log of five is equal to one to find a particular solution. So, letβs integrate six π to the power of π₯ plus two.

We begin by recalling that the integral of π to the power of π₯ is π to the power of π₯. Then, the integral of two is two π₯. Now, of course, this is an indefinite integral. So, we need that constant of integration. And we can say that our function is defined as π of π₯ equals six π to the power of π₯ plus two π₯ plus some constant π. Now, we said that π of the natural log of five is equal to one. In other words, when π₯ is equal to the natural log of five, π of π₯ is equal to one. So, we replace π of π₯ with one. And then, each time we see an π₯ in our equation, we replace that with the natural log of five.

So, one is equal to six π to the power of the natural log of five plus two times the natural log of five plus π. Of course, π to the power of the natural log of five is just five. So, six π to the power of the natural log of five is six times five, which is 30. We now see that we can solve for π by subtracting 30 and two times the natural log of five from both sides of our equation. And when we do, we find that π is equal to negative 29 minus two times the natural log of five.

We then go back to the general solution for π of π₯, and we replace π with negative 29 minus two times the natural log of five. And so, we found π of π₯. Itβs six π to the power of π₯ plus two π₯ minus 29 minus two times the natural log of five.