Integration question?

Can I integrate a function with substitution when the limits are outside the range of the substitution function? eg.

\displaystyle\int^{2}_{-2} x\ dx = ...

(obviously you wouldn't use a substitution here), letting

u = cos x

. As obviously

cos x \not={ -2 or 2}

Reply 1

lizard54142

Original post by JackB003

Can I integrate a function with substitution when the limits are outside the range of the substitution function? eg.

\displaystyle\int^{2}_{-2} x\ dx = ...

(obviously you wouldn't use a substitution here), letting

u = cos x

. As obviously

cos x \not={ -2 or 2}

I think you meant let

x = \cos u

? Generally no, you cannot, but in certain situations you can (such as when a function has an asymptote).

Reply 2

Gome44

Original post by lizard54142

I think you meant let

x = \cos u

? Generally no, you cannot, but in certain situations you can (such as when a function has an asymptote).

No, you can put u=cos(x), so you're essentially using x=arccos(u).

Edit: Nvm, I see why you said that :smile:

Reply 3

lizard54142

Original post by Gome44

No, you can put u=cos(x), so you're essentially using x=arccos(u)

I believe the OP wanted a situation where the substitution was undefined for those particular values.

Reply 4

JackB003

If you'd like a proper example the use this:

\displaystyle\int^{2}_{-2} \frac{1}{\sqrt{1-x^2}}\ dx

where

x = sin u

(edited 8 years ago)

Reply 5

lizard54142

Original post by JackB003

If you'd like a proper example the use this:

\displaystyle\int^{2}_{-2} \frac{1}{\sqrt{1-x^2}}\ dx

where

x = sin u

This does not have a "real" solution, you'd be delving into the world of complex numbers.

Reply 6

davros

Original post by JackB003

If you'd like a proper example the use this:

\displaystyle\int^{2}_{-2} \frac{1}{\sqrt{1-x^2}}\ dx

where

x = sin u

What do you think that integral represents?

Can you draw a graph of the function you're integrating over the range of integration? :biggrin:

Reply 7

JackB003

Original post by davros

What do you think that integral represents?

Can you draw a graph of the function you're integrating over the range of integration? :biggrin:

Yeah I did that afterwards and know I can't think of a valid example... :/

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Integration question?

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