Calculus integration help?

[k3n]

You can try trig substitution maybe.

haha

 

[HN]

I didn't pass my AP exam for calculus A/B but did pass it for B/C so got credit for both A/B and B/C

 

[DrPizza]

A fairly powerful technique when you have square roots is to use trig substitutions, or alternatively, hyperbolic trig substitutions.

First, combine the terms inside the square root, so you have the square root of (4x+1)/(4x)

If you break it into a two square roots, and simplify the root in the denominator, you have
sqrt(4x+1) / 2sqrt(x)

Let 2sqrt(x) = tan(theta) *(I have some reservations with this step; can explain later, but haven't thought domain issues through completely)
Then,
4x = tan²(theta)
1/sqrt(x) dx = sec²(theta) dtheta
How convenient, since the integral has a sqrt(x) in the denominator.

So, using these substitutions, you end up with
1/2 integral sqrt(tan²(theta)+1) sec²(theta) dtheta

A quick trig simplification and it's
1/2 integral (sec³(theta)) dtheta

It's fairly simple** to integrate that using integration by parts. Alternately, you can use a reduction formula for integrating powers of sec(theta).

So, you get 1/2[ 1/2 sec(theta)tan(theta) + 1/2 ln (abs (sec(theta)+tan(theta)))]

Factor out the 1/2s and 1/4 [sec(theta)tan(theta) + ln|sec(theta)+tan(theta)|
We already know that tan(theta) = 2sqrt(x) (That was our original substitution.)
And, by either applying a trig identity, else employing some pythagorean theorem, we can deduce that sec(theta)= sqrt(1+4x)

So, finally, 1/4[2sqrt(1+4x)sqrt(x) + ln( sqrt(1+4x)+2sqrt(x) )

I verified this answer numerically for one definite integral - I used a TI 83+ to calculate the integral from 1 to 8. (=7.253)

And, using this formula for the definite integral yielded 8.73245-1.47353 = 7.253



**Note: on the integral of secant cubed, wikipedia notes "is a frequent and challenging indefinite integral of elementary calculus." I say bullshit, but they have the footnote for it.

 

[Aaviel]

Who ARE you?

Lurker. Only name I go by is this one.. I was actually around like 2 or 3 years before I made this username too...

 

[boomhower]

I certainly don't miss that crap. To think of the math courses I went through and the career I chose. Good grief.

 

[exdeath]

I verified this answer numerically for one definite integral - I used a TI 83+ to calculate the integral from 1 to 8. (=7.253)

And, using this formula for the definite integral yielded 8.73245-1.47353 = 7.253

Hah. I always did this on every test question (verify numerically with a few test cases).

 

[NutBucket]

I certainly don't miss that crap. To think of the math courses I went through and the career I chose. Good grief.

I went through a ton of math in undergrad...some in grad. I haven't touched the stuff in years.

 

[exdeath]

I went through a ton of math in undergrad...some in grad. I haven't touched the stuff in years.

I have but mostly numerical integration of a data stream in software programming (Euler, Runge-Kutta, etc). For example, physics simulation , a = sum(F)/m and integrate over an interval to obtain velocity and position in real time.

Real life data is almost never available in neat algebraic functions that can be solved symbolically.

If I thought hard enough, I probably use it all the time without realizing. I just take for granted the fundamentals instilled into me as a result of understanding the coursework, even if some of the more obscene math course material doesn't apply literally.