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## Integral Calculus (2017 edition)

### Course: Integral Calculus (2017 edition)>Unit 6

Lesson 6: 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? • 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? • Some practice exercises will be added for u-substituiton, trig substituiton and other integrals? Thanks. • 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? • 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) • 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 kind of thought you could integrate something like ∫(cosx)^2 dx by parts, as ∫(cosx)(cosx)dx. But when I do that, I come up with a tautological statement. Am I doing something wrong, or can it just not be solved that way?
(1 vote) • 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). 