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I'm trying to compute the following integral:

$$\int e^{3x}\cos2x \;dx$$

Now I'm about to use the integration by parts. Suppose that I do not know what is the integral of $\displaystyle \int \cos2x\; dx$. Is it a good practice to write it like this:

$$\int \stackrel{f}{e^{3x}}\cdot \stackrel{g'}{\cos2x} \;dx=e^{3x} \cdot\int \cos2x\; dx -\int 3e^{3x}\cdot \left( \int \cos2x\; dx \right) \; dx$$

And then compute the desired antiderivatives that I don't know later?

Red Banana
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    Well, if you do not know $\int \cos 2x dx$ is, I don't see any reason why you want to transform the LHS to the RHS, using integration by part. –  Jul 06 '15 at 14:25
  • @JohnMa I'm supposing I don't know it for a while. But I'll compute it a little later. Do you think this presents a big problem? – Red Banana Jul 06 '15 at 14:32
  • The problem is that you might not known how to move on... What is the next step then? –  Jul 06 '15 at 14:33

2 Answers2

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Instead of using integration by parts, you may just assume that: $$\int e^{3x}\cos(2x)\,dx = e^{3x}\left( A\sin(2x)+B\cos(2x)\right)+C $$ then find $A,B$ through differentiation. That is not much different from noticing that: $$\int e^{3x}\cos(2x)\,dx = \text{Re}\int e^{(3+2i)x}\,dx = \text{Re}\left(\frac{1}{3+2i}\,e^{(3+2i)x}+C\right) $$ from which $A=\frac{2}{13},B=\frac{3}{13}$ follows.

Jack D'Aurizio
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That's sort of OK...but I like to think of the integral sign and the "dx" as being like open- and close-parentheses: they have to match. Furthermore, you're not allowed to have nested integrals with the same variable. [These are just my own rules for avoiding writing nonsense, not some axioms of mathematics! But if you write something that violates these rules, it probably won't make sense.]

To fix the "matching parens" problem, you'd need to change to $$ \int \stackrel{f}{e^{3x}}\cdot \stackrel{g'}{\cos2x} \;dx=e^{3x} \cdot\int \cos2x\; dx -\int 3e^{3x}\cdot \left( \int \cos2x\; dx \right) \; dx $$ but then you'd have two nested integrals with the same variable of integration. You CAN write this as something like

$$ \int \stackrel{f}{e^{3x}}\cdot \stackrel{g'}{\cos2x} \;dx=e^{3x} \cdot\int \cos2x\; dx -\int 3e^{3x}\cdot \left( \int^x \cos2t\; dt \right) \; dx $$

which says "compute the antiderivative with respect to $t$, and then evaluate at $t = x$". I've always disliked that notation, but it works for some people.

The problem here is that we use the integral sign to denote two very different things: an antiderivative and a definite integral. These are, of course, very closely related -- by the Fundamental Theorem of Calculus -- but it's still a notational nightmare. If you use $\mathcal A_x (Q )$ to mean "the set of all antidervatives of the function $x \mapsto Q$ (where $Q$ is some expression), then your formula would look like $$ \mathcal A_x( \stackrel{f}{e^{3x}}\cdot \stackrel{g'}{\cos2x} ) = e^{3x} \cdot \mathcal A_x(\cos2x) -\mathcal A_x( 3e^{3x}\cdot \mathcal A_x(\cos2x)) $$ which would at least be notationally consistent, although not as suggestive as the integral form.

By the way, the "equality" here is one of sets -- the set of all antiderivatives on the left is equal to the set of functions you get on the right. Or, to put it differently, if you want $\mathcal A_x$ to mean "an antiderivative" rather than "the set of all antiderivatives", then you need to add "$+ C$" on one side or the other.

John Hughes
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  • What is the advantage of denoting $\frac12 \sin(2x)$ by a definite integral $\cos(2t)$? Or is it not the goal to evaluate the original integral? – zoli Jul 06 '15 at 14:30
  • The asked us to *pretend" that he didn't yet know how to find the antiderivative of $\cos(2x)$, and wanted some way to denote that thing in the middle stages of integration by parts. As an example of this situation,if you had to integrate $\cos({x^2}) \frac{x^2+1}{x^3 - 1}$, you might say "I'll use parts, differentiating the first and integrating the second", while not yet knowing the antiderivative of the second, but certain that it exists, because you've done lots of partial-fractions integrals in your life. (I doubt this'll lead to a solution, but it's an approach one might take.) – John Hughes Jul 06 '15 at 18:23
  • In particular: if the part you do know doesn't simplify nicely, you might decide not to invest the effort in the as-yet-unknown indefinite integral -- you could consider this a kind of exploratory approach to the "garden of forking paths" that is the process of antidifferentiation. – John Hughes Jul 06 '15 at 18:34