Question

Can anyone help me to do this definite integration?

Answer 1

\[\int\limits_{-1}^{3/2}\left| xsin Pix \right|\]

Answer 2

no,can anyone help me to do this?

Answer 3

How are you stuck?

Answer 4

how to split it into two halves?

Answer 5

split what?

Answer 6

Do you want to know how to solve \[
\int_a^b |f(x)|~dx
\]?

Answer 7

yeah

Answer 8

One way is to find the roots of \(f(x)\).
Then you'll be able to figure out the intervals where \(|f(x)|=f(x)\) and \(|f(x)|=-f(x)\).

Answer 9

And you integrate on those intervals separately.

Answer 10

@wio That is what i am asking?

Answer 11

You don't know how to find roots?

Answer 12

@wio can you help in this case i couldnot find the roots

Answer 13

Notice xsin(pix) is positive between x = -1 and x = 1
this is because both x and sin(pix) have same polarity in this interval

between 1 and 3/2 only sin(pix) is negative so xsin(pix) will be negative

so you may split the integral at x = 1

Answer 14

Okay so \(f(x) = x\sin(\pi x)\)
When either factor is a root, then the whole thing is a root.
\(x\) has a root only at \(x=0\).
\(\sin(\pi x)\) has a root at \(\pi x=\pi n\) meaning \(x=n\) where \(n\) is some integer.

Answer 15

We are looking at the interval from \(-1\) to \(3/2\), which contains the integers \(-1,0,1\).

Answer 16

So our sub-intervals will be \([-1,0]\), \([0,1]\), and \([1,3/2]\).

Answer 17

two integrals will do : \[\int\limits_{-1}^{3/2}\left| x\sin (\pi x) \right| = \left| \int\limits_{-1}^{1}x\sin (\pi x) \, dx\right| + \left| \int\limits_{1}^{3/2}x\sin (\pi x) \,dx\right| \]

Answer 18

I suggest you work the integral of xsin(pix) and plug it in above to avoid working by parts two times

Answer 19

Yeah ,i understood.But can u say why you told that x has a root at x=0

Answer 20

@wio

Answer 21

Yeah ,i understood.But can u say why you told that x has a root at x=0

Answer 22

Because, hypothetically even if \(\sin(\pi x)\) didn't have a root at \(x=0\), we'd have to consider it as a root.