2

It is very hard for me sometimes to find a right method to do an integration, for example, whether to use by parts method or substitution. I always find the questions I was given on an exam very hard to do without spending much time experimenting on it (I have to skip otherwise I will run out of time). An example question would be to integrate an arcsin(x).

So I wonder if there's any site that provides solutions for all different kinds of integration or something that will help me to recall instantly the method I need to use to integrate an equation? or any advice would be appreciate!!

greentea123
  • 23
  • although I do have experience, by what I know, the method is to try all ways. Sry.(Don't you think differentiation is much more easier?) –  May 12 '14 at 12:52
  • The site is here, Math SE. :) – Tunk-Fey May 12 '14 at 12:52
  • 2
    This is probably not the answer you are looking for but i recomend that you solve a lot of exercises. if you practice enough you'll see that it's going to be A LOT easier to decide (on the majority of cases anyway) which method to use. i'm telling you this as someone who learned this things not so long ago :) – essay May 12 '14 at 12:53
  • 2
    Related: Some users are mind bogglingly skilled at integration. How did they get there? – MJD May 12 '14 at 13:02
  • I liked real analysis during college, but solving all those integrals in first year calculus was possibly my least favorite part of all my math studies. In other words, they gave us the same hazing 35 years ago. The advice in the previous comments is very good; I hope it helps. If you choose to go on to other math classes, they get a lot more fun when you get past this. – David K May 12 '14 at 13:08
  • @greentea123 book problems will generally tell you which to use. For problems in the wild you are going to have to experiment. – bobbym May 12 '14 at 13:18
  • you can also try this site for particular integrals you want to solve quickly: https://www.wolframalpha.com . in the pro version it shows you step-by-step solution, and you can get a trial account with this option for free – mm-aops May 12 '14 at 13:33
  • Consider to check out the non-standard analysis if you have difficulties about the concepts of integrals and derivatives themselves, about how they are defined for example. The NSA makes the treatment of these topics with the more intuitive infinitesimal approach rigorous and IMHO, it is more understandable compared to classic sigma- epsilon analysis. But in order to solve given integrals in an exam, you still need to remember the fundamental rules. More skill comes with practice in that particular area. – Ufuk Can Bicici May 12 '14 at 13:45
  • See also http://math.stackexchange.com/questions/20578/why-is-integration-so-much-harder-than-differentiation. – lhf May 12 '14 at 13:59
  • See this answer in another question. – Felix Marin May 12 '14 at 14:42
  • 1
    I suggest looking through some older calculus texts at a (good) public library or a college library, books dating roughly from the 1950s through the 1970s. Most of the more complete (really means: thickest, with most number of pages) calculus texts from these decades will have extensive treatments of integration techniques. Much of this material has been de-emphasized or even omitted in more recent texts, the more recent the more de-emphasized and/or omitted. When you find a text you like, you can probably buy it for a few dollars somewhere on the internet. – Dave L. Renfro May 12 '14 at 15:38

1 Answers1

3

1) You need to know the tables of derivatives/primitives of elementary functions by heart, flawlessly. (Recall that powers yield powers, polynomials yield polynomials, exponential and $\sin/\cos$ yield themselves.)

2) Recognize factors of the integrand that you can integrate, i.e. write $f$ as $u'.v$.

2a) In some cases, $v$ can be expressed as a function of $u$. For example, $x.e^{x^2}=(x^2)'.e^{x^2}$ hints the substitution $u=x^2$. (Unessential constant omitted.)

2b) In other cases, it may appear that $u.v'$ would be somewhat simpler to integrate: try integration by parts. For example, in $x.\cos x$, you can integrate both $x$ or $\cos x$. Deriving the other factor will give $\sin x$ and $1$ respectively, resulting in integrands $x^2.\sin x$ (worse than before) or $\sin x$ (immediate).

2c) If you can't find an integrable factor, try this trick: $\int f(x)dx=\int x'f(x)dx=xf(x)-\int xf'(x)dx$.

In the case of $\arcsin x$, you won't find a suitable primitive in the tables (but note that the derivative is $\frac1{\sqrt{1-x^2}}$). You don't see an integrable factor either. Then observe that by using rule 2c, you will get rid of the $\arcsin$ !

Now you have to solve $\int \frac x{\sqrt{1-x^2}}dx$. You can integrate $x$ as $x^2$, a candidate for the $u$ function. But an even better $u$ candidate is $1-x^2$, as appearing under the radical, also having $x$ for derivative. It will yield $\int \frac{dx}{\sqrt x}$, found in the tables.

For this case, another strategy can be attempted: get rid of the annoying $\arcsin$ function from the beginning by subsitituting $x$ with $\sin x$. You integrand will turn to the form $\arcsin(\sin x).\cos x=x.\cos x$.