You can’t evaluate this integral using any of the techniques that you have studied so far, so use
the calculator to evaluate the integral numerically.
Remember this: Use your calculator to evaluate integrals whenever you can.
You should get approximately 1.090. The AP Exam always expects you to round to three
decimal places.
- B The Second Fundamental Theorem of Calculus tells us how to find the derivative of an
integral. It says that f(t) dt = f(u) , where c is a constant and u is a function of x.
Here we can use the theorem to get
sin^2 t dt = (sin^2 (x^2 ))(2x), or 2x sin^2 (x^2 )
- C Here we use implicit differentiation to find .
2 xy + x^2 + 2y = 0
Now we plug y = 1 into the original equation to find its corresponding x-values.
x^2 + 1 = 5
x^2 = 4
x = ±2
Now plug in the x- and y-values to find the value of .
For y = 1 and x = 2, we get
2(2)(1) + (2)^2 + 2(1) = 0
