Different Ways of Integration?

Question

Are there different methods to integrate indefinite integrals?

Sometimes it takes too long to think about the question and sometimes I just don't know how to do it.

So instead of integration by substitution, parts, trigonometric substitution, or partial fractions, what else can I use?

Answer 1

[edit: Kyle, on ∫ tan x - it's a question of what's worth memorising separately. For me the integral of tan x is obvious enough that it's not worth memorising; it's a simple application of the rule ∫f'(x)dx/f(x) = ln|f(x)| + c, which is itself a special case of substitution. Since it's not hard to recognise sin x as being -d/dx cos x I don't find it necessary to memorise this case; it's easy enough to derive mentally that it wouldn't slow me down. But obviously this is a highly individual decision.]

Unfortunately there is no general method of integration. If there was, we probably wouldn't teach you all the other ones (unless the general method was REALLY HARD).

Offhand I can't think of any additional methods beyond those you list, but there are a couple of tricks in relation to integration by parts which I feel are worth mentioning:

1. In some cases you can integrate a function f(x) more easily by writing the integral as ∫1.f(x) dx = x f(x) - ∫x.f'(x) dx. The classic example of this is integrating ln x: ∫ln x = x ln x - ∫x (1/x) dx = x ln x - x + c.

2. Sometimes you can use integration by parts to return the original integral multiplied by some constant as part of the RHS. Then you can bring that over to the left and group by the integral to get the result. You often need to use integration by parts mutiple times to achieve this. Used particularly in conjunction with trig functions, e.g.
∫ cos x e^x dx: use integration by parts, differentiate the trig part and integrate the exponential:
= cos x e^x - ∫ (-sin x) e^x dx
And again:
= cos x e^x + sin x e^x - ∫ (cos x) e^x dx
Now bring the integral part over to the left to get
2 ∫ (cos x) e^x dx = cos x e^x + sin x e^x
=> ∫cos x e^x dx = e^x (cos x + sin x) / 2.

One last trick that's worth mentioning is for certain infinite integrals; the classic example is ∫(0 to ∞) e^(-x^2) dx. Call this integral K, then we have
K^2 = (∫(0 to ∞) e^(-x^2) dx) (∫(0 to ∞) e^(-y^2) dy)
= ∫(0 to ∞) ∫(0 to ∞) e^-(x^2 + y^2) dy dx
Now we change to polar coordinates:
= ∫(0 to ∞) ∫(0 to π/2) e^(-r^2) r dθ dr
= π/2 ∫(0 to ∞) e^(-z) (dz/2) substituting r^2 = z
= π/4 [-e^-z] [0 to ∞]
= π/4 (0 + 1)
So K = √π / 2.

Note that this trick doesn't work on finite or indefinite integrals, because the shape of a finite rectangular region can't be easily expressed in polar coordinates.

Answer 2

the first answer is sweet. once you're making y = a million/(2x) into y = (a million/2) * x^(-a million) --ensure this step is genuine in case you go with to attain understanding Your crucial might want to then change into extremely straightforward. y = (a million/2) * ln|x| + C because you could't strengthen the -a million skill to change into the 0 skill, it truly is why the crucial is ln|x|. continually positioned fractional variables of their exponential type: eg. a million/x = x^(-a million) this may continually make integrating a lot a lot less annoying to comprehend and calculate.

Answer 3

The only other one I can think of is my personal favorite:

Memorization.

Example:
int(tan(x)) = -ln|cos(x)| + C

How do I know that? Memorization. Can I derive it? Sure. Do I want to derive it on every exam/problem I do? No.