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# Indefinite integrals of sin(x), cos(x), and eˣ

AP.CALC:
FUN‑6 (EU)
,
FUN‑6.C (LO)
,
FUN‑6.C.1 (EK)
,
FUN‑6.C.2 (EK)

## Video transcript

thought I would do a few more examples of taking anti-derivatives just so we feel comfortable taking anti-derivatives all of the functions the the basic functions that we know how to take the derivatives of and on top of that I just want to make it clear that it doesn't always have to be functions of X here we have a function of T and we're taking the antiderivative with respect to T and so you would not write a DX here that is not the notation you'll see why when we focus on definite integrals so what's the antiderivative of this business right over here well it's going to be the same thing as the antiderivative of sine of T it's going to be the antiderivative of sine of T or the indefinite integral of sine of T plus the indefinite integral or the antiderivative of cosine of T plus the antiderivative of cosine of T so let's think about what these anti derivatives are and we already know a little bit about taking the derivatives of trig functions we know that the derivative with respect to T of cosine of T is equal to negative negative sine of T so if we want a sine of T here we just have to take the derivative of negative cosine T take the derivative of negative cosine T then we get positive sine of T derivative with respect to T if cosine T is negative sine of T we have the negative out front it becomes positive sine of T so the antiderivative of sine of T the antiderivative is negative cosine of T so this is going to be equal to negative cosine of T and then what's the antiderivative of cosine of T well we already know that the derivative with respect to T of sine of T is equal to cosine of T so cosine of T is antiderivative is just sine of T so plus sine of T and we're done we found the antiderivative now let's tackle this now we don't have a t where where we're taking the ant we're taking the indefinite integral with respect to actually this is a mistake this should be with respect to a let me clean this up this should be a DA if we're taking this with respect to T that we would treat all of these things as just constants but I don't want to confuse you right now let's move let me make it clear this is going to be d that's what we are integrating or taking the antiderivative with respect to so what is this going to be equal to well let's once again we can rewrite it as the sum of integrals this is the indefinite integral of e to the a DA so this one right over here plus I'll do it in green plus the indefinite integral or the antiderivative of 1 over a da now what is the antiderivative of e to the a well we already know a little bit about Exponential's the derivative with respect to X of e to the X is equal to e to the X that's one of the reasons why e and the exponential function in general is so amazing and if we just replaced a with X or X with a you get the derivative with respect to a of e to the a is equal to e to the a so the antiderivative here the derivative of e to the ADA is e to the a the antiderivative is going to be e to the a and maybe you could shift it by some type of a constant oh and let me make let me not forget I have to put my constant right over here I could have a constant factor so let me always important remember the constant so you have a constant factor right over here never forget that I almost did so once again over here what's the antiderivative of e to the a it is e to the a what's the antiderivative of 1 over a well we've seen that in the last video it is going to be the natural log of the absolute value of a and then we want to have the most general antiderivative so there could be a constant factor out here as well and we are done we found the antiderivative of both of these both of these expressions
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