Posted by **Eric** on Wednesday, May 15, 2013 at 10:51am.

Find dy/dx and d2y/dx2 if y= definite integral sign where a= 1 and b= 3x

1/(t^2+t+1) dt

how do i even start. do i integrate and then plug in a and b? plz help.

- integral -
**Steve**, Wednesday, May 15, 2013 at 11:19am
dy/dx = 1/(x^2+x+1) * 3

- integral -
**Eric**, Wednesday, May 15, 2013 at 11:30am
can you explain how you got this. thanks

- integral -
**Steve**, Wednesday, May 15, 2013 at 11:35am
excuse me? Didn't we just go through this about differentiating under the integral sign?

- integral -
**Eric**, Wednesday, May 15, 2013 at 11:38am
but i don't get how you can just replace the variable t with x so easily. i don't get how you did that. and how does the 3 come into the picture for dy/dx.

- integral -
**Steve**, Wednesday, May 15, 2013 at 12:04pm
In general (see the wikipedia article for proof),

∫[a(x),b(x)] f(t) dt

= f(b(x)) db/dx - f(a(x)) da/dx

Since we have a(x) = 1, da/dx = 0 and we are left with

∫[a(x),b(x)] f(t) dt

= f(b(x)) db/dx

b(x) = 3x, so db/dx = 3

So, oops. I made a mistake. It should be

dy/dx = 1/((3x)^2+3x+1) * 3

Good catch! :-)

- integral -
**Eric**, Wednesday, May 15, 2013 at 12:12pm
Thank you very much! I really didn't understand the form but now I do. :)

Also, would d2y/dx2 be:

-9(6x+1)/((9x^2+3x+1)^2)

I just want to make sure that d2y/dx2 is asking for the second derivative?

- integral -
**Steve**, Wednesday, May 15, 2013 at 12:42pm
haven't checked your work, but yes, they want y"

- integral -
**Eric**, Wednesday, May 15, 2013 at 1:49pm
Oh okay thank you!

## Answer This Question

## Related Questions

- Integral - Find dy/dx and d2y/dx2 if y= definite integral sign where a= 1 and b...
- calculus (please with steps and explanations) - consider the function f that is ...
- Calculus (urgent help) - consider the function f that is continuous on the ...
- calculus - consider the function f that is continuous on the interval [-5,5] and...
- Calculus II/III - A. Find the integral of the following function. Integral of (x...
- Math - Find the integrals. (show steps) (integral sign) xe^(4x^2) I think this ...
- Calc, - Find the value of the definite integral: int= integral sign int((x + 2...
- Calc - If we know that the definite integral from -6 to -3 of f(x) equals 6, the...
- Calc. Help Please. - Find the value of the definite integral using sums, not ...
- calculus - There are four integrals: 1) definite integral x/(1+x^4)dx b/w ...

More Related Questions