Section The fundamental Theorem of Calculus:
Use Part I of the fundamental Theorem to compute each integral exactly.
4
| 4 / 1 + x^2 dx
0
To compute the integral ∫(4 / (1 + x^2)) dx using Part I of the Fundamental Theorem of Calculus, we need to find an antiderivative of the function 4 / (1 + x^2) first.
Let's denote the antiderivative of 4 / (1 + x^2) as F(x). According to the Fundamental Theorem of Calculus, if F(x) is an antiderivative of f(x), then ∫(f(x)) dx from a to b is equal to F(b) - F(a).
Now, let's find F(x) by recognizing that 4 / (1 + x^2) is similar to the derivative of the arctan function.
The derivative of arctan(x) is 1 / (1 + x^2). So, if we multiply this derivative by 4, we obtain the function 4 / (1 + x^2).
Therefore, F(x) = 4 * arctan(x) + C, where C is the constant of integration.
Now, we can use Part I of the Fundamental Theorem of Calculus to compute the definite integral from 0 to 4:
∫[0,4] (4 / (1 + x^2)) dx = F(4) - F(0) = [4 * arctan(4) + C] - [4 * arctan(0) + C]
= 4 * arctan(4) - 4 * arctan(0)
= 4 * (arctan(4) - 0)
= 4 * arctan(4)
Therefore, the exact value of the integral ∫[0,4] (4 / (1 + x^2)) dx is 4 * arctan(4).