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### Course: Integral Calculus>Unit 1

Lesson 16: Trigonometric substitution

# Long trig sub problem

More practice with a hairy trig sub problem. Created by Sal Khan.

## Want to join the conversation?

• correct me if im wrong but didnt he get the problem wrong because he stopped carrying over the +C?
• No-It's kind of understood to be there (during the problem), but your final answer HAS to have the constant of integration, or C.
• So, Sal shows sample problems for the forms (a^2 - x^2) as well as (a^2 +x^2); however, I did not see any problems with the form (x^2 - a^2). One of the practice problems I was given took this form, and I thought that hyperbolic trig substitution would be appropriate since we can use the identity cosh^2(theta) - 1 = sinh^2(theta). I arrived at a reduced answer to the problem in terms of inverse hyperbolic trig functions; however all of the multiple choice answers were in terms circular trig functions. Is there an easy substitution I'm missing?
• Here is another solution, but with an example (it might even be a question from Khan).
This is my very first LaTeX document - so, 'scuse any weirdness.
http://bajasound.com/khan/khan1.pdf
• Is integration just a process of trial and error and remembering types or are there any things to look out for and do to bring a expression into a expression that can be easily integrated ?
As there are no formulas like quotient rule or chain rule for it as in differentiation if a complicated expression to integrate is given how should it be approached ?
• There are tips and tricks for integration. For example, the power rule is (I think) the simplest integration rule. It is really the reverse of the power rule for derivatives: d/dx (x^n) = nx^(n-1)
The power rule for integrals says: ∫ x^n dx = ( x^(n+1) ) / (n+1)
There are also methods of integration like trig sub, u sub, integration by parts, partial fraction decomp...
Knowing what methods to use when just requires a lot of practice. You can probably find practice problems if you search google (Khan Academy does not have integration practice modules).
• I got the same solution as the video from doing it manually, but how come wolfram's integration calculator got (1/2)(x-3)(sqrt(-x^2+6x-5)-2arcsin((3-x)/2) instead?
• There are different ways of solving these problems. You can get very different-looking, but mathematically equivalent answers depending on how you solved the problem. This is especially the case when you have trigonometric functions involved.
• Some practice exercises will be added for u-substituiton, trig substituiton and other integrals? Thanks.
• There is always more videos than exercises, because they are produced much faster (as Sal said in one of his videos). But I am sure your patience will pay off, its just a matter of time before we get exercises of all kinds :D
• Can you just solve this and these types of problems with normal substitution? You could just set 6x - x^2 - 5 = u and go on from there. Trig substitution seems unnecessary and long. Or am I wrong and you can't use regular substitution?
• Try out the substitution you've suggested to see if it works. :)
The challenge you'll run in to is with du and dx.
• Sir, you have replaced (x-3)/2 with sin theta. Does that not alter the values x can have? Replacing it with sin function makes x lie between 1 and 5 only.. Is that acceptable??
(1 vote)
• Look at the original problem. ∫√(6x-x²-5)dx Any value inside the square root that's less than 0 gives you a problem already. So 6x-x²-5 ≥ 0 already meaning 1 ≤ x ≤ 5. So replacing it with sin doesn't actually change anything.
• sir I am student of XI standard from Calcutta,India...and I have been watching your videos since class 8....
I am really a great fan of yours......

Actually I have a question that has been disturbing for a couple of weeks and i really can't solve it by own...
Is it possible to integrate this function: x^x (with respect to dx)..?

Actually I have a question
• I entered this into wolframalpha.com and confirmed that the integral cannot be written with standard mathematical functions. The integral can be written only in terms of an infinite series, and the series is quite complicated.

Have a blessed, wonderful day!
• Why doesn't he use u substitution in this problem? I find that it is easier than to do that (x-3/2)^2 business (sorry if that expression is incorrect).
• Some problems can be solved using both u substitution and trig substitution, but in my experience u substitutions are often easily identifiable, you can usually see a function and its derivative in the problem or at least manipulate it to get the function and its derivative. (In most cases)

So even though you may find a u substitution easier, in this case that might not be true or even possible as far as I can see. So look at the problem and think what you can possibly do to change it up a bit, for a second order polynomial such as this you can complete the square, and often completing the square leads to a constant and a (function)^2. Once you have that you can apply a trig substitution because that allready looks like the standard forms a^2-x^2 in the above example.

Doing enough examples of these kinds of problem is the only way to get to know what they want you to do, In 90% of the cases the way the question is asked, hints at how you should answer it.