Evaluate improper integral: Discontinuous integrand

[HugoAng]

Homework Statement

:

Evaluate: ∫_-2¹⁴ (1+X)^-1/4[/B]

Homework Equations

∫_a^b f(x)dx =
lim ∫_a^t f(x)dx
t→b^-

And

∫_a^b f(x)dx =
lim ∫_t^b f(x)dx
t→a⁺

The Attempt at a Solution

So far what I have done is:
(-2,-1)∪(-1,14)

Thus I set up two integrals and chose one to evaluate: [/B]
∫_-2^-1 dx/((1+x)^1/4 + ∫_-1¹⁴ dx/(1+x)^1/4

The second integral gave me a finite answer of 4/3(15)^3/4 but the first one always leads me to an imaginary unit. From this point I understand WHY this is happening, and that led me to think the problem may have a typo but the thing is my professor told us that there is an answer. Also, no one in my class got this question correct so it was turned into homework. Any help will be appreciated. Thanks!

 

[GFauxPas]

Why is it a bad thing that you're getting a complex number as an answer?

 

[HugoAng]

Why is it a bad thing that you're getting a complex number as an answer?

Well because as of now I do not what it means in regards to the concept. We have not studied that yet.

 

[Mark44]

Homework Statement

:
[/B]
Evaluate: ∫_-2¹⁴ (1+X)^-1/4

The Attempt at a Solution

So far what I have done is:
(-2,-1)∪(-1,14)

Thus I set up two integrals and chose one to evaluate: [/B]
∫_-2^-1 dx/((1+x)^-1/4 + ∫_-1¹⁴ dx/(1+x)^-1/4

The second integral gave me a finite answer of 4/3(15)^3/4 but the first one always leads me to an imaginary unit. From this point I understand WHY this is happening, and that led me to think the problem may have a typo but the thing is my professor told us that there is an answer. Also, no one in my class got this question correct so it was turned into homework. Any help will be appreciated. Thanks!

I'm confused as to what the integral is. In the problem statement you have this:
$$\int_{-2}^{14} (1 + x)^{-1/4}dx$$
but in your work below there, you have integrands that look like this:
$$\frac{dx}{(1 + x)^{-1/4}}$$

 

[Ray Vickson]

Homework Statement

:

Evaluate: ∫_-2¹⁴ (1+X)^-1/4[/B]

Homework Equations

∫_a^b f(x)dx =
lim ∫_a^t f(x)dx
t→b^-

And

∫_a^b f(x)dx =
lim ∫_t^b f(x)dx
t→a⁺

The Attempt at a Solution

So far what I have done is:
(-2,-1)∪(-1,14)

Thus I set up two integrals and chose one to evaluate: [/B]
∫_-2^-1 dx/((1+x)^-1/4 + ∫_-1¹⁴ dx/(1+x)^-1/4

The second integral gave me a finite answer of 4/3(15)^3/4 but the first one always leads me to an imaginary unit. From this point I understand WHY this is happening, and that led me to think the problem may have a typo but the thing is my professor told us that there is an answer. Also, no one in my class got this question correct so it was turned into homework. Any help will be appreciated. Thanks!

For $x < -1$ (with $|x| > 1$) the integrand $f(x) = (1+x)^{-1/4}$ equals$(-1)^{-1/4}(|x|-1)^{-1/4}$. We have $(-1)^{-1/4} = e^{-i \pi/4} = (1-i)/\sqrt{2}$, where $i = \sqrt{-1}$ is the imaginary unit. Since $f$ is complex-valued for $x < -1$ its integral from -2 to -1 is also complex-valued.

 

[HugoAng]

I'm confused as to what the integral is. In the problem statement you have this:
$$\int_{-2}^{14} (1 + x)^{-1/4}dx$$
but in your work below there, you have integrands that look like this:
$$\frac{dx}{(1 + x)^{-1/4}}$$

Oops! Let me fix that real quick! I'm barely getting used to the forums.
It should be:
$$\frac{dx}{(1 + x)^{1/4}}$$[/QUOTE]