Thread: Integration technique help, please!

What technique should I use to solve this one?

int(16+x^2)^(-3/2)

I tried a few techniques, including partial fractions, but I couldn't work it out. Can anybody help me?

Originally Posted by riverjib

What technique should I use to solve this one?

int(16+x^2)^(-3/2)

I tried a few techniques, including partial fractions, but I couldn't work it out. Can anybody help me?

Start with the substitution:
x = 4*tan(t)
and note that
1 + tan^2(t) = sec^2(t)

-Dan

Originally Posted by riverjib

What technique should I use to solve this one?

int(16+x^2)^(-3/2)

I tried a few techniques, including partial fractions, but I couldn't work it out. Can anybody help me?

I agree with topsquark, trig substitution seems like the way to go.

you have int(16+x^2)^(-3/2) = int((4)^2+x^2)^(-3/2) = int[1/(sqrt((4)^2 + x^2)^3].

trig substitution says, if your integral contains sqrt(a^2 + x^2), where a is a constant, you use the sustitution, x = atan@ and the identity 1 + tan^2@ = sex^2@

Originally Posted by Jhevon

1 + tan^2@ = sex^2@

Uhh.....Heh heh heh heh. He said "sex" Beavis.

-Dan

Originally Posted by topsquark

Uhh.....Heh heh heh heh. He said "sex" Beavis.

-Dan

haha, yea, i did. it should be sec(x), haha, thats what trying to type fast gets you

Thank you both! That makes perfect sense.

I started by using the substitution x=4tan@, but how do i handle the fact that a^2+x^2 is cubed?? none of the trig substitution problems we've done have added that kind of complication.

Originally Posted by riverjib

Int[(16+x^2)^(-3/2)dx]

Let x = 4*tan(t) ==> dx = 4*sec^2(t) dt

Then
Int[(16+x^2)^(-3/2)dx] =
Int[(16+16{tan^2(t)})^(-3/2) * 4sec^2(t) dt]

= 4*(16)^(-3/2)*Int[(1 + tan^2(t))^(-3/2) sec^2(t) dt]

= 4*(1/64)*Int[(sec^2(t))^(-3/2) sec^2(t) dt]

= (4/64)*Int[sec^2(t)/sec^3(t) dt]

= (1/16)*Int[1/sec(t) dt]

= (1/16)*Int[cos(t) dt]

= (1/16)*sin(t)

Now, t = tan^{-1}(x/4), thus

Int[(16+x^2)^(-3/2)dx] = (1/16)*sin[tan^{-1}(x/4)]
(plus some constant of integration.)

Now, imagine a right triangle with legs x and 4, then the hypotenuse will be sqrt(x^2 + 16). We have an angle t (which is tan^{-1}(x/4)) across from the side of length x. Then the sine of this angle will be: x/sqrt(x^2 + 16). So
sin[tan^{-1}(x/4)] = x/sqrt(x^2 + 16)

So finally:
Int[(16+x^2)^(-3/2)dx] = (1/16)*x/sqrt(x^2 + 16)

(Note: I finally found the error in my original post. This is now correct.)

-Dan

Thanks Dan! I read over your solution, gave it a few hours, and then worked it out on my own. Now I really understand every trig substitution problem in the book. I'm ready for my exam :-)

I really appreciate all your help.

Originally Posted by riverjib

Thanks Dan! I read over your solution, gave it a few hours, and then worked it out on my own. Now I really understand every trig substitution problem in the book. I'm ready for my exam :-)

I really appreciate all your help.

You are very welcome! For the record I found the error in my post and have corrected it.

-Dan