# Thread: Tricky definite integral problem.

1. ## Tricky definite integral problem.

I am struggling in calculating a tricky integral

$\int^{\x}_a (t)}dt$

Please show me the step and intuition of calculating this definite integral...

Thanks!

2. What exactly do you mean? I think your notation is a bit messed up. Here's a standard definite integral. You can double-click to see the code required:

$\displaystyle{\int_{0}^{1}t^{2}\,dt=\frac{1}{3}.}$

3. Sorry, I was figuring out how to type integral symbol here... The upper limit of the integral is x, which is not displayed on the screen

To make it clear, the lower limit of the integral is a, the upper limit of the integral is x and the variable of integration is t. Thanks.

4. So you want

$\displaystyle{\int_{a}^{x}t\,dt},$ right?

5. Yes Sorry for the trouble...

6. No problem. Problem definition usually consumes a very large part of problem-solving, as it should.

You can solve this integral using the Fundamental Theorem of the Calculus. So, what do you get for the antiderivative?

7. So the answer is just F(x)???

If consider about the first fundamental theorem of calculus, then F(x) is the only result... I was also considering about this solution but I thought the answer should not be so simple

8. Well, technically, F(t). But you should know how to compute that antiderivative, and give the closed-form expression for it. What is the antiderivative of t with respect to t?

9. I got the answer now... !

t^2/2!!!!

10. No, there are no x's. When you compute an antiderivative, you completely ignore the limits (in this case, a and x) of the integral. The antiderivative depends entirely on the integrand (in this case, t), and the variable of integration (also t). Try again: what do you get?

11. got it!!! I posted my result before your new post XD Thankssssss aaaa lllloooootttt!!!

12. This is indeed a tricky yet very very interesting question!!!

13. So the answer is... ?

14. t^2/2 + C. I edited the post which contained the wrong result!

15. Ah, I see. So now you plug in the limits as per the Fundamental Theorem, and you get what?

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