Integration

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April 13th 2012, 12:18 PM
qwerty31

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Integration

These few questions have been bugging me for hours, help would be appreciated
April 13th 2012, 12:35 PM
Siron

Re: Integration

1. What's the lower bound? Use the goniometric substitution $t=\tan(u)$
2. Use integration by parts.
April 14th 2012, 03:00 AM
qwerty31

Re: Integration

Quote:

Originally Posted by Siron

1. What's the lower bound? Use the goniometric substitution $t=\tan(u)$
2. Use integration by parts.

Thank you :) How do you know when to use t = tanu?

Also would you be able to give me an indication of what to do for this one:

Integrate 9t/(9t^2 + 6t +5)
April 14th 2012, 05:39 AM
Siron

Re: Integration

Quote:

Originally Posted by qwerty31

Thank you :) How do you know when to use t = tanu?

Because with that subsitution you can 'eliminate' the root form which is easier to work with. (Have you solved the integral?)

Quote:

Originally Posted by qwerty31

Also would you be able to give me an indication of what to do for this one:
Integrate 9t/(9t^2 + 6t +5)

$\int \frac{9t}{9t^2+6t+5}dt = \frac{1}{2} \int \frac{(18t+6)-6}{9t^2+6t+5}dt = \frac{1}{2} \left(\int \frac{18t+6}{9t^2+6t+5}dt - \int \frac{6}{9t^2+6t+5}dt\right)$

I guess you can take it from here. To solve $\int \frac{6}{9t^2+6t+5}dt$ write $9t^2+6t+5=(3t+1)^2+4$ (because $D<0$ )
April 14th 2012, 05:47 AM
HallsofIvy

Re: Integration

$sin^2u+ cos^2u= 1$ so, dividing both sides by $cos^2 u$ , $tan^2u+ 1= sec^2u$ . Because of that the substitution $x= tan u$ changes $\sqrt{1+ x^2}$ to $\sqrt{1+ tan^2 u}= \sqrt{sec^2 u}= sec u$ . Generally speaking, any time you have a square root of a quadratic, you should consider a trig substitution. (Some people prefer hyperbolic functions which satisfy similar identities.)

As for your new question, first complete the square: $9t^2+ 6t+ 5= 9(t^2+ (2/3)t+ 1/9- 1/9)+ 5)= 9(t+ 1/3)^2+ 4$ .
Now the substitution x= t+ 1/3 changes to integrand to $\frac{9u- 3}{9u^2+ 4}= \frac{u}{u^2+ (2/3)^2}+ (4/9)\frac{1}{(3u/2)^2+ 1}$