Use the Evaluation Theorem to find the exact value of the integral
�ç^6 2 2x+1dx
1) What is the antiderivative?
2)What is theupper and lower limit?
3) Give final answer.
To find the exact value of the integral ∫(2x+1)dx with upper and lower limits of a=2 and b=6, we can use the Evaluation Theorem. This theorem states that if F(x) is the antiderivative of f(x), then the definite integral of f(x) from a to b is given by F(b) - F(a).
Now let's go through the process step by step:
1) Find the antiderivative:
To find the antiderivative of 2x+1, we integrate each term separately. The antiderivative of 2x is x^2, and the antiderivative of 1 is x. So the antiderivative of 2x+1 is (x^2 + x).
2) Determine the upper and lower limits:
In this case, the upper limit is b=6 and the lower limit is a=2. These limits define the range over which we are integrating the function.
3) Use the Evaluation Theorem to find the value of the integral:
According to the Evaluation Theorem, the integral ∫(2x+1)dx from a to b is equal to the antiderivative evaluated at the upper limit minus the antiderivative evaluated at the lower limit. Therefore, the exact value of the integral is:
∫[a to b] (2x+1)dx = [(x^2 + x)] [evaluated from a=2 to b=6]
= [(6^2 + 6)] - [(2^2 + 2)]
= [36 + 6] - [4 + 2]
= 42 - 6
= 36
So, the exact value of the integral ∫[2 to 6] (2x+1)dx is 36.