- #1

- 59

- 0

## Homework Statement

[tex]

\int_{-3}^{3}|t|e^{-jwt}dt

[/tex]

## The Attempt at a Solution

I am not sure if I need to break this into two regions due to the abs value...

- Thread starter Xkaliber
- Start date

- #1

- 59

- 0

[tex]

\int_{-3}^{3}|t|e^{-jwt}dt

[/tex]

I am not sure if I need to break this into two regions due to the abs value...

- #2

CompuChip

Science Advisor

Homework Helper

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[tex]\int_{-3}^3 |t| e^{-j \omega t} \, dt = \int_{0}^3 t e^{-j \omega t} \, dt - \int_{-3}^0 t e^{-j \omega t} \, dt[/tex]

and then you can solve both integrals with a trick (write the integrand as a derivative w.r.t. omega, for example).

Alternatively, you can use Euler's identity to write the integral as

[tex]\int_{-3}^3 |t| \cos(\omega t) \, dt + j \int_{-3}^3 |t| \sin(\omega t) \, dt[/tex]

and use (anti)-symmetries to reduce the problem before taking care of the absolute value.

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