Evaluate the definite integrals �ç[-2,0,(e^2+1),x] + �ç[0,1,2 cos(x),]
To evaluate the definite integrals, let's go through each integral step by step.
1. First integral: ∫[e^2+1, x]
To evaluate this integral, we need to find the antiderivative of the integrand 1 with respect to x. The antiderivative of 1 is x. Now, we can compute the definite integral by evaluating the antiderivative at the upper and lower limits of integration:
∫[e^2+1, x] = [x] evaluated from e^2+1 to x = x - (e^2+1)
2. Second integral: ∫[2 cos(x), 0, 1]
Here, we have the integrand 2 cos(x). To evaluate this integral, we need to find the antiderivative of cos(x). The antiderivative of cos(x) is sin(x). Now, we can compute the definite integral by evaluating the antiderivative at the upper and lower limits of integration:
∫[2 cos(x), 0, 1] = [2 sin(x)] evaluated from 0 to 1 = 2 sin(1) - 2 sin(0) = 2 sin(1)
Finally, we can add the results of the two integrals:
∫[-2,0, (e^2+1),x] + ∫[0,1,2 cos(x)] = (x - (e^2+1)) + (2 sin(1))
So, the evaluated expression for the definite integrals is (x - (e^2+1)) + (2 sin(1)).