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First of all, it is very important to simply learn some common integrations by rote (e.g trig functions, exponentials, polynomials, 1/x). If you know these well, it will often be easy to spot which technique to use when integrating a seemingly difficult function.
If you are in a situation where you have to integrate a function comprised of two different types of function, such as f(x)=xe^x, integration by parts can be useful (i.e substitute u for one part of the function and v' for the other, such that the you now have to integrate uv' with solution uv - integral(vu'dx)). It is often a good idea when choosing what to use as u and v to choose as u whatever simplifies the most when you differentiate. For example, in the above function I would make u=x and v'=e^x so that u'=1 and v=e^x. Then the integral of vu' is simply the integral of e^x which is quite simple.
Unless you are given case like this, integration by inspection (just by looking at it and using your known results) is always possible but can sometimes be hard to spot. If this is the case, it can be useful to use a clever substitution to help you. (Have a look at my solution to the integral of f(x)=x(1-x)^6 for an example of thissee more
Every sentence in Latin can be broken up into smaller, simple parts. The first step is always to identify and perhaps even highlight all the words which agree with each other in case, number and gender. Often times, if nouns and adjectives come from different declensions or one or more are irregular, it can be difficult to spot what agrees with what and mistakes can be made. For this reason, it is imperative that you learn your grammar tables and vocab.
Next, look for your main verb. This will usually be at the end of the sentence and tends to be indicative (not subjunctive) - again, learning verb tables thoroughly is vital. Generally, if a verb is subjunctive, that is a good sign that it is part of a subordinate clause.
Once you have completed these two quick steps, you then have to piece together all of the parts of the sentence in a way which makes sense. Usually there will only be one correct solution so if you think there is more than one possibility, consider two options:
1) Do your two solutions have precisely the same meaning (be very rigorous in determining the answer)? If so, then either is fine. If not;
2) Revise your grouping and specification of words. The chances are that you have misidentified the case/number/gender of a word or group of words or you have mistranslated the tense/mood of a verb.see more
For a question like this, it would be far too time-consuming to expand the bracket and then multiply through by x. It might then seem that integration by parts is the optimal solution, but this is actually not necessary.
If you make the substitution u=1-x then you have du=-dx. Then the integral of x(1-x)^6 with respect to x is equivalent to the integral of -(1-u)u^6 with respect to u. By multiplying through, we have u^7-u^6 which is not difficult to integrate.
An important thing to remember in this is always to remember to find dx in terms of du - do not assume that dx=du because it rarely does. Finally, make sure you sub x back into your final answer.
If you are given a definite integral where you need a substitution, always remember to change your limits appropriately. There is then no need to sub x into your integral at any point in the workingsee more