Equations (4) and (5) lead to different formulas for the integ...

Question

Equations (4) and (5) lead to different formulas for the integral of arctan x:

a. ∫ arctan x dx = x arctan x - ln sec(arctan x) + C [Eq. (4)]

b. ∫ arctan x dx = x arctan x - ln √(1 + x²) + C [Eq. (5)]

Can both integrations be correct? Explain.

Accepted Answer

Hello there. Today we are going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Let i equal the interval of inverse tangent of 2 x dx. Using integration by parts, the integral can be written as the integral of the inverse tangent of 2 x dx is equal to x multiplied by inverse tangent of 2 x. Minus 1/2 multiplied by the natural log of m x plus c for all real x, what is the function m x such that m x is greater than 0? OK. So it appears for this particular problem, we're asked to take all the information that is provided to us, and we're asked to use this information to determine what is the function m x. So what does m x equal, such that m x is greater than 0. So now that we know that we're ultimately trying to solve for m x. Our first step that we need to take is we need to take a moment to understand what we're trying to do to solve for this problem. So based on our prior knowledge and based on what the problem tells us, in order to find the function m of X, we will need to recall and apply the integration by parts method. By treating the integrand as a product of the inverse tangent function and one. After differentiating the inverse trigometric part and integrating the constant, we will then need to evaluate the resulting integral by recalling and using a standard substitution method in order to match the logaririththic form provided in the problem itself. And that's a quick overview of what we're, what we are going to do in order to solve this problem. So with that in mind, let's break it up into steps. So let's take our overview and break it up into steps. So our first step is we're going to apply the integration by parts formula or integration by parts method by using the formula, and the integration by parts formula states that the integral of uDV is equal to u multiplied by v minus the integral of v multiplied by du. And let us say that for this problem, when we choose our variables that you will be equal to inverse tangent of 2 X. So that will mean we need to differentiate both sides with respect to X. And when we do that, we will be able to write that du by dx will be equal to 1 divided by 1 + 2 x to the power of 2 multiplied by 2, which will mean that duu will be equal to 2 divided by 1 plus 4 x to the power of 2. DX and let us state that DV will be equal to dx, which will mean that the integral of DV is equal to the integral of dx, which will mean as a result that v will be equal to x. So when we plug these variables into our integration by parts formula, we will be able to write that the integral of inverse tangent of 2 x dx will be equal to x multiplied by inverse tangent of 2 x minus the integral of 2 x divided by 1 + 4 multiplied by x to the power of 2 dx. Fantastic. Our second step now is we need to solve for the remaining integral, and in order to solve for the integral of. 2 X divided by. 1 Plus 4 x to the power of 2 dx, we will now need to, in order to solve for this this once again, this specific integral, we will need to use our U substitution method and let us use the variable w to help us avoid confusion, since we've already used U and V because traditionally you'll use like U or V. But to make sure we keep everything straight, let's use a different variable. So let us state for this problem that, or for this stage, I should say, that W will be equal to 1 + 4 multiplied by x to the power of 2. That will mean that DW by DX will be equal to. 8 X, which will mean that X multiplied by DX will be equal to 1/8 DW. So now we need to plug these variables, rather substitute these variables back into our integral. So the integral of 2 X divided by 1 + 4 multiplied by X to the power of 2. The X will be equal to 2 multiplied by the integral of 1 divided by w multiplied by 1/8 DW, which will be equal to 1/4 multiplied by the integral of 1 divided by w. Dw, which will mean that 1/4 multiplied by the integral of 1 divided by w dw will be equal to 1/4 multiplied by the natural log of the absolute value of w, which will be equal to 1/4 multiplied by the natural log of 1 + 4 multiplied by x to the power of 2. Fantastic. So our 3rd step is we now need to combine our results and compare. So now we need to substitute back into our integration by parts result. So we need to take the integral of. Inverse tangent of 2 x dx is equal to x multiplied by inverse tangent of 2 x minus 1/4 multiplied by the natural log of parentheses of 1 + 4 multiplied by x to the power of 2. Plus C where C denotes some constant variable, that's what capital C represents some constant variable. So with that in mind. Now the prom states that the form is, so we need to note that the prom itself states that the form is X multiplied by inverse tangent of 2 X. 1/2 multiplied by the natural log of M of X. Plus C. So we need to note that in order to make our result match this form, we will need to recall and use the logarithm power rule which states that k multiplied by the natural log of a. Will be equal to the natural log of a to the power of k. So with that in mind, we can go ahead and write that. 1/4 multiplied by the natural log of 1 + 4 multiplied by x 2 will be equal to 1/2 multiplied by parentheses of 1/2 multiplied by the natural log of 1 + 4 multiplied by x to the 2. Which will be equal to 1/2 multiplied by the natural log of parentheses of 1 + 4 multiplied by x 2 1/22. Which will give us 1/2 multiplied by the natural log of the square root of 1 + 4 multiplied by x to the power of 2. Fantastic, we are on a roll. We are on a roll here, everybody. So now our fourth and final step is we need to identify m of x. So when we compare the two expressions, we'll have 1/2 multiplied by the natural log of the square root of 1 + 4 multiplied by x 2, which is equal to 1/2 multiplied by the natural log of m x. Therefore, when we rearrange this expression to isolate and solve for m of x, we will find that our function m x will be simply equal to the square root of 1 + 4 multiplied by x to the power of 2. And that's it. That is our final answer. Hooray, we did it. So going back to look at the problem itself, we could safely say without a doubt that our final answer for our function m of x such that m of x is greater than 0 will be once again so our final answer without a doubt. We have to be. M X is equal to the square root of 1 + 4 multiplied by X to the power of 2, and that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Equations (4) and (5) lead to different formulas for the integral of arctan x:a. ∫ arctan x dx = x arctan x - ln sec(arctan x) + C [Eq. (4)]b. ∫ arctan x dx = x arctan x - ln √(1 + x²) + C [Eq. (5)]Can both integrations be correct? Explain.

Watch next

Equations (4) and (5) lead to different formulas for the integral of arctan x:
a. ∫ arctan x dx = x arctan x - ln sec(arctan x) + C [Eq. (4)]
b. ∫ arctan x dx = x arctan x - ln √(1 + x²) + C [Eq. (5)]
Can both integrations be correct? Explain.