Question

Re-write the following improper integrals using limit notation?

Answer 1

The following improper integrals using limit notation is:

`\[ \int_(1)^(\infty) (1)/(x^2) \,dx = \lim_{{a \to \infty}} \int_(1)^(a) (1)/(x^2) \,dx \]`

Step-by-step explanation:

Improper integrals arise when the interval of integration extends to infinity or involves a discontinuity. The given integral
`\(\int_(1)^(\infty) (1)/(x^2) \,dx\)` is improper because it extends to infinity. To rewrite it using limit notation, we introduce a variable
`\(a\) and rewrite the integral as \(\int_(1)^(a) (1)/(x^2) \,dx\).` This new integral has a finite upper limit, and then we take the limit as
`\(a\)`approaches infinity to recover the original improper integral.

In more detail, by introducing the limit notation
`\(\lim_{{a \to \infty}}\),` we express the idea that we are letting the upper limit of integration
`(\(a\))` go to infinity. The limit ensures that we approach the improper integral in a controlled manner, allowing us to deal with infinity in a mathematically rigorous way. This technique is essential in handling integrals that involve unbounded regions, providing a precise mathematical representation of the process as we extend the interval to infinity.

Answer 2

There are the following improper integrals by using limit notation;

1.
`\( \lim_{{a \to 0^+}} \int_(a)^(B) (1)/(x) \, dx \)`

2.
`\( \lim_{{c \to -\infty}} \int_(c)^(1) (e^t)/(t^2) \, dt \)`

3.
`\( \lim_{{d \to \infty}} \int_(2)^(d) (√(u))/(u^2 - 1) \, du \)`

Step-by-step explanation:

1. For the integral
`\( \int_(0)^(\infty) (1)/(x) \, dx \)`, the limit notation represents the improper integral. By setting a as the lower limit approaching zero from the positive side, the integral becomes
`\( \lim_{{a \to 0^+}} \int_(a)^(B) (1)/(x) \, dx \),` where B is the upper bound, which here is infinity.

2. The integral
`\( \int_(-\infty)^(1) (e^t)/(t^2) \, dt \)` is expressed using limit notation as
`\( \lim_{{c \to -\infty}} \int_(c)^(1) (e^t)/(t^2) \, dt \)`. Here, the limit c tends towards negative infinity, representing the lower bound approaching negative infinity while the upper bound remains constant at 1.

3. For
`\( \int_(2)^(\infty) (√(u))/(u^2 - 1) \, du \)`, the limit notation is
`\( \lim_{{d \to \infty}} \int_(2)^(d) (√(u))/(u^2 - 1) \, du \)`, where d tends towards infinity as the upper limit while 2 remains the lower limit.

Here is complete question;

"Rewrite the following improper integrals using limit notation, clearly expressing the limits of integration and the integrands for each case:"

1.
`\( \int_(0)^(\infty) (1)/(x) \, dx \)`

2.
`\( \int_(-\infty)^(1) (e^t)/(t^2) \, dt \)`

3.
`\( \int_(2)^(\infty) (√(u))/(u^2 - 1) \, du \)`"