How to Integrate x.e^x and x^2.e^x

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Integrating x.e^x Using a Predetermined Result

A simplified predetermined result can be used to integrate x.e^x to save your time and effort during exams and homework. This result can be memorized easily and its application will make the solution shorter. It also avoids the cumbersome integration by parts when solving for ∫ ( x.e^x ) dx.

Predetermined result for integration of x.e^x
My engineering textbook

where a = constant.

Note: This is the same result you will get when integrating x.e^ax manually by parts.

Integrals of the type x.e^x are often encountered in engineering subjects, particularly when evaluating coefficients for exponential Fourier series. General Interpretation can be to apply a u.v integration by parts to solve for a solution however if you adopt this approach solution can become considerably long and time consuming.

Solved Example for integration of x.e^x

Let's demonstrate this with the help of a solved example:

Q: ∫( x.e^-2x)

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Integration of x^2 . e^x Using Predetermined Result

Integration of x².e^xcan be done in the same way as x.e^xusing a similar predetermined result, stated below.

Predetermined result for integration of x^2.e^ax.
Own effort.

Where, a = constant.

Note: This is also the same result you will get when integrating x².e^ax manually by parts.

Solved Example for integration of x^2.e^x

Let's demonstrate this with the help of an example which finds c_nfor complex Fourier series of x²using our predetermined result.

Q: ∫( x².e^-jwnx)

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.