Finding an Indefinite Integral In Exercises 15- 36 , find the indefinite integral and check the result by differentiation.

$\int\left(x^{5}+1\right) d x$

$\frac{1}{6} x^{6}+x+c$

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Numerade Educator

Missouri State University

Baylor University

University of Nottingham

Oh, eso One thing that we could do to make this problem. Ah, little bit easier as we can actually split this up. It's a to enter growth. And the reason we could do that is because we simply just have the the X five plus one with no power attached to it. So what, This will look like this This right here So we could say the integral of X of the five DX plus the integral off one d X. So now that just made the problem a lot more easier. Because now, instead of us having this combined into growth, we have just to smaller and the growth too soft. So let's start with the very first integral, which is our X to the fifth power. So if you remember any time we have any time we want to take the integral off some verbal raises, some constant end. The result will always be that verbal race to that power plus one, Tom. And then we would also divide by that new power. So close one. Okay, so we apply that to this first integral, our xrm, and then our five is our innocence. That's the constant So what we should see is that we have X to the power five close one all over five plus one. And do not forget, since this is an indefinite integral, uh, there's always gonna be plus some constant that we just don't know. And then if we go to the second and a growth, this one is very simple eso In this case, we have some constant and the X the solution to this one who just simply be that constant times x. So since we're adding these to inter girls, you can say plus and our case, our constant is only one. So that's going to give us one x. Don't forget, This is also in a definite integral so plus C So now we can simplify our result. So now we have X to the power of six and then this is also divided by six. So we can just simply say one over six. If you remember from algebra way, multiply this fraction. This is 1/6 times X over one because there's always that visible one under. So you have 1/6 times. Excellent six and then we still have that X from our second indefinite, Integral. So that's another ex and which you may notice right here is that we have to seize. This isn't really important. I mean, it's important. We do want to include the fact that we have a C. We do wanna like notified that we have a constant, but we don't have to put to sea. And the reason that been is that we're not necessarily looking for what this constant is exactly. And if you just think critically for a second 22 constants is still a constant. And since we're not looking for the particular constant, we could just leave this to see, So that is your final answer 1/6 times X to the power of six plus x plus our constant