Let g be the function given by g(x) = the integral from 0 to x sin(t^2) for -1 < or equal to x < or equal to 3. find the interval where the function g(x) is increasing.

If you're using the app, try seeing this answer through your browser:  

_______________

•  According to what is given,



•  Now, differentiate g by using the Fundamental Theorem of Calculus:



•  g is increasing in the interval where g'(x) is positive. So now, just solve this inequality:



•  The sine function is positive for angles that lie either in the first or the second quadrant. So,



•  The inequality above involves only non-negative terms. So, the sign of the inequality keeps the same for the square root of those terms:



•  Checking the intersection between the interval we just found above and the domain of g:

Notice that



which implies that



Therefore,

g is increasing on the interval  

I hope this helps. =)

Tags:  derivative fundamental theorem of calculus increasing interval differential integral calculus