Evaluate the following integral: ∫ 1 4 ( 2 x + 1 ) d x \int_1^4\!\left(2x+1\right)\,dx ∫ 1 4 ( 2 x + 1 ) d x

So in this problem, we're asked to evaluate the following integral. Now notice we have a definite integral that goes from one to four, and it's of this function right here. Now let's see how we can go about solving this. Now recall that the fundamental theorem of calculus gives us a way to deal with these definite integrals. And our first step is going to be to find the antiderivative of this function right here. Now I noticed that I have the integral of two x plus one, so I can actually integrate each of these separately using the rules that I know for some indifference when it comes to in antiderivatives or definite integrals in this case too. So what I can do is start with this two x here. Now you what I can do is use the power rule here on this two x, and that's going to give me two x squared over two, and they're going to have plus and then the antiderivative of one which is just x. I can go ahead and cancel these twos right here, and it's going to give me x squared plus x, and then this is all going to be bounded from one to four. So this right here is the antiderivative we're dealing with, and this is the bounds we need to apply, so let's go ahead and do that. Now first off we need to plug in our high bound of four. So we're going to have x squared, which is gonna be four squared plus four, and then we're going to take this whole thing and subtract off the low bound of one. So doing this we're going to have one squared plus one, and then that's gonna be what we combine here. So let's go ahead and see how we can simplify things. Four squared is 16, then we add four to that we're going to get 20. Here we're going to get this minus then one squared plus one which comes out to two. 20 minus two is 18, giving us 18 as our result, and that is the solution to this problem. So as you can see, when it comes to dealing with these definite integrals, all you need to do is find the antiderivative of this function and then apply the bounds. And using that fundamental theorem of calculus allows us to evaluate any of these types of problems that we see. So hope you found this video helpful, and let's move on.

8. Definite Integrals

Fundamental Theorem of Calculus

Business Calculus

8. Definite Integrals

Fundamental Theorem of Calculus

Struggling with Business Calculus?

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Multiple Choice

Evaluate the following integral:
$\int_1^4\!\left(2x+1\right)\,dx$

Verified step by step guidance

Step 1: Recognize that the integral ∫₁⁴ (2x + 1) dx is a definite integral, which means we will evaluate the antiderivative of the function (2x + 1) and then apply the limits of integration (from x = 1 to x = 4).

Step 2: Find the antiderivative of the function (2x + 1). The antiderivative of 2x is x², and the antiderivative of 1 is x. Therefore, the antiderivative of (2x + 1) is F(x) = x² + x.

Step 3: Apply the Fundamental Theorem of Calculus, which states that for a definite integral ∫ₐᵇ f(x) dx, we evaluate F(b) - F(a), where F(x) is the antiderivative of f(x). Here, F(x) = x² + x, a = 1, and b = 4.

Step 4: Substitute the upper limit (b = 4) into the antiderivative F(x). This gives F(4) = 4² + 4.

Step 5: Substitute the lower limit (a = 1) into the antiderivative F(x). This gives F(1) = 1² + 1. Finally, compute F(4) - F(1) to find the value of the definite integral.