Few Problems I can not figure out

[mathpat]

1. [ 0,7] f(x)dx = 8 , [1,7] f(x)dx = -3 , [0,1] f(x)dx=?

2. Evaluate the definite Integral [-2, 4] x^2(x^3+8)^2 dx

3. [2 ,x] 1/(1+t^4) dt. Find F' (x) and F' (2)



Thanks

 

[stapel]

1. [ 0,7] f(x)dx = 8 , [1,7] f(x)dx = -3 , [0,1] f(x)dx=?

This exercise is quite cryptic. What is the meaning of the intervals (?) in the square brackets? What are you supposed to be doing with these various expressions? My only guess is that maybe the book means something like this:

\(\displaystyle \mbox{If }\, \int_0^7\, f(x)\, dx\, =\, 8\, \mbox{ and }\, \int_1^7\, f(x)\, dx\, =\, -3,\,\mbox{find the value of }\, \int_0^1\, f(x)\, dx.\)

But that's a lot of assuming. Please reply with confirmation or corrections.

2. Evaluate the definite Integral [-2, 4] x^2(x^3+8)^2 dx

I will guess that the square-bracketed interval is meant to indicate the limits of the definite integral. Where are you stuck in doing the integration?

3. [2 ,x] 1/(1+t^4) dt. Find F' (x) and F' (2)

How does the integral (?) relate to "F"?

 

[mathpat]

yea I'm sorry about that. You are making the correct assumptions. I did not know how to enter those problems in that form.

For the second problem it states to find F' (x) and F' (2).

For the third problem, I do not know what to substitute.

For the first problem, i get 11?

 

[DrPhil]

yea I'm sorry about that. You are making the correct assumptions. I did not know how to enter those problems in that form.

For the second problem it states to find F' (x) and F' (2).

For the third problem, I do not know what to substitute.

For the first problem, i get 11?

The form
Int[a,b](. . . )dx
works pretty well - we who post here a lot use LaTeX, but that is not easy to learn.

1st problem correct, but without the "?" symbol.

2. This is not the one that asks for F(x) and F'(x). The definite integral is not a function of x.
Multiply out the factors of the integrand, and you will have a polynomial in x that can be integrated term-by-term.

3. \(\displaystyle \displaystyle F(x)\ =\ \int_2^x \dfrac{1}{1+t^4}\ dt \)

This one is a function of x, because x is the upper limit. Fortunately you are not asked to find F(x), which would be a major undertaking. Just F'(x), which is easily found using the Fundamental Theorem of Calculus.