Video: Integrating Reciprocal Trigonometric Functions

For the given function 𝑓(π‘₯) = ln (1 + 2π‘₯), find a power series representation for 𝑓 by integrating the power series for 𝑓′.

03:42

Video Transcript

For the given function 𝑓 of π‘₯ equals the natural log of one plus two π‘₯, find a power series representation for 𝑓 by integrating the power series for 𝑓 prime.

We don’t have a nice power series expansion for the function 𝑓 of π‘₯ equals the natural log of one plus two π‘₯. But we should notice that the derivative of 𝑓, 𝑓 prime of π‘₯, is two over one plus two π‘₯. So we’re going to start with an equation that we’ve seen before. That is, one over one minus π‘₯ is equal to one plus π‘₯ plus π‘₯ squared plus π‘₯ cubed, and so on. And we write this as the sum of π‘₯ to the 𝑛th power for values of 𝑛 between zero and ∞.

We write our derivative as two times one over one plus two π‘₯. And then we’re going to use the equation we’ve seen before, replacing π‘₯ with negative two π‘₯. And this is because we want it in the form one minus something. And one minus negative two π‘₯ gives us the one plus two π‘₯ we’re after. We can therefore use this equation to write one over one plus two π‘₯ as one plus negative two π‘₯ plus negative two π‘₯ squared plus negative two π‘₯ cubed, and so on. Which can of course be written as the sum of negative two π‘₯ to the 𝑛th power for values of 𝑛 between zero and ∞.

𝑓 prime of π‘₯ is two times one over one plus two π‘₯. So that’s two times the sum of negative two π‘₯ to the 𝑛th power for values of 𝑛 between zero and ∞. Two is independent of 𝑛. So we’re going to take it inside the sum. And we’re going to rewrite negative two π‘₯ as two times negative one times π‘₯.

We then distribute this exponent across each term. And we see that we have the sum of two times two to the 𝑛th power times negative one to the 𝑛th power times π‘₯ to the 𝑛th power. Two times two to the 𝑛th power can be written as two to the power of 𝑛 plus one. And now we have our expression for 𝑓 prime of π‘₯.

We’re trying to find a power series representation for 𝑓. So we recall that we can achieve this by integrating our expression for 𝑓 prime of π‘₯. And we can do so by integrating each individual term in the series. That’s called term-by-term integration. Here that’s the integral of the sum of two to the power of 𝑛 plus one times negative one to the 𝑛th power times π‘₯ to the 𝑛th power between 𝑛 equals zero and ∞ with respect to π‘₯.

And then of course since we’re dealing with a power series, we can write this as the sum of the integrals. So how are we going to integrate two to the power of 𝑛 plus one times negative one to the 𝑛th power times π‘₯ to the 𝑛th power? Well, no matter the value of 𝑛, two to the power of 𝑛 plus one times negative one to the 𝑛th power is a constant. This means we can take it outside of the integral and focus on integrating π‘₯ to the 𝑛th power itself.

Now when we integrate π‘₯ to the 𝑛th power, we know that 𝑛 is positive. So we simply add one to the exponent and then divide by that new number. So we get π‘₯ to the power of 𝑛 plus one over 𝑛 plus one. And of course we had that constant of integration 𝐢, which is outside the summation. So how do we find that constant of integration 𝐢?

Well, let’s go back to 𝑓 of π‘₯. We’re told that it’s equal to the natural log of one plus two π‘₯. And we know that if we let π‘₯ be equal to zero, we get quite a nice value for 𝑓 of zero. It’s the natural log of one plus two times zero, which is the natural log of one, which is of course zero. By replacing 𝑓 of π‘₯ with zero and π‘₯ with zero, we see that zero is equal to the sum of two to the power of 𝑛 plus one times negative one to the 𝑛th power plus zero to the power of 𝑛 plus one over zero plus one plus 𝐢.

Now zero to the power of 𝑛 plus one over zero plus one is always zero. And so we have the sum of zeros, which is zero. And we find the constant of integration itself is zero. And so by integrating the power series for 𝑓 prime, we found a power series representation for 𝑓. It’s the sum of two to the power of 𝑛 plus one times negative one to the 𝑛th power times π‘₯ to the power of 𝑛 plus one over 𝑛 plus one.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.