Integral, At what value of t does the local max of f(t) occur?

Question

Given



Integral (a=0.b=t) [(x^2+11x+18)/(1+cos^2(x)]dx

At what value of does the local max of occur?

Answer 1

Think of the integral as a function, F(t), of t. According to the second fundamental theorem of integral calculus, the derivative of F(t) with respect to t is

F' = (t² + 11 t +18) /( 1 + cos²(t) )

This derivative will have zeros where the numerator is zero. It is easy to find these value of t by factoring. The values are t = -9 and t = -2 . A plot of F' shows that the slope of F'(t) is negative at t = -9 and positive at t = -2. Thus the second derivative of F is negative at t =-9 and positive at t = -2. Thus the local maximum of F is at t = -9, while a local minimum occurs at t = -2 .