Solving Integration Problem: Int 0-2 (1+9x^4)^1/2 dx

[BoldKnight399]

integral from 0 to 2 of (1+9x^4)^1/2 dx

I was thinking that I could use a trig substitution to solve for it so it would be:
[2]\int[/0] [9(1/9 +x^4)]^1/2
so [2]\int[/0]3[(1/9+x^4)]^1/2

my problem is that I can understand that my a would b 1/3, and that it should be atantheta, but what I can't seem to get is that it is an X^4. I understand that it is (x^2)^2, but again, how do I do it because I only know how to deal with it when it is x^2. If anyone has any suggestions how to deal with this problem, that would be great.

 

[cragar]

ok here goes nothing
i start out by x=1/(sqrt(3))tan(u) dx=1/sqrt(3)sec(u)^2
so then w have 1/(sqrt3)*sqrt(1+tan(u)^4)*sec(u)^2du
then i am using the formula for reducing powers for tan(u)^2
which is tan(u)^2=(1-cos(2u))/(1+cos(2u)) then substituting
this in for tan^2 and then foiling this
and then getting a common denominator with the one inside the radical and combining like terms to give me
sqrt(2+2cos(2x)/(1+2cos(2x)+cos(2x)^2)
then i factor the top and bottom
can u see how it factors 2(1+u)/(u+1)^2 u=cos(2x)
leaving us with 2/(cos(2x)+1) which if you notice this is the inverse of the formula for reducing powers for cos^2) so this is sec^2(u) so this simplifies beautifully to
sqrt(sec^2)(sec^2)1/(sqrt(3)
which becomes sec^3(u) so we then integrate sec^3 which we know how to do
which becomes 1/2(secu)tan(u)+1/2ln|secu+tanu| and our 1/sqrt(3) out front
then from our original substitution x=1/sqrt(3)tanu we simply draw our little triangle
and find the sec and tan of it and back substitute . i hope this is right ,

 

[BoldKnight399]

wait...how did you get a cube root?

 

[Mark44]

It's not a cube root -- sqrt(3) is the square root of 3. I think that's what you're referring to.

 

[BoldKnight399]

I am still lost though as to how you got that. And why is it 1 over that?

 

[Mark44]

I think what cragar did was to use a trig substitution. I'll let him/her jump back in and correct me if the details aren't right.

Draw a right triangle with one acute angle labelled u. The opposite side is 3x^2 and the adjacent side is 1. The hypotenuse is sqrt(1 + 9x^4). From this triangle, tan u = 3x^2, so x = sqrt(tan u)/sqrt(3). Also, sec^2(u)du = 6x dx. With these relationships you can rewrite the original integral so that it is in terms of u and du.

 

[BoldKnight399]

ok, that makes sense but how do you plug it back it? Do you plug it back in so that the 3x^2 is substituted by tanu, or is just the x substituted by tan^(1/2)/root3? Thats where I understood that it should have been a trig sub, but I didn't know how to do it with a x^4

 

[Mark44]

You substitute based on the the relations I gave in post #6, that are based on the right triangle I described. You can get everything from that.

 

[BoldKnight399]

oh, ok. I get it now, Thank you