Question:

Integrate the function: \(x^2e^x\)

Updated On: Oct 4, 2023
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Solution and Explanation

Let I = ∫x2ex dx

Taking \(x^2\) as first function and ex as second function and integrating by parts, we obtain

I= x2∫ex dx - ∫{\((\frac {d}{dx}x^2)\)∫exdx} dx

I = x2ex - ∫2x.exdx

I = x2ex-2∫x.ex dx

Again integrating by parts,we obtain

I =x2ex - 2[x.∫exdx - ∫{\((\frac {d}{dx}x)\).∫exdx} dx]

I = x2ex - 2[xex-∫exdx]

I = x2ex - 2[xex - ex]

I = x2ex - 2xex + 2ex + C

I = ex(x2 - 2x + 2) + C

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Concepts Used:

Integration by Partial Fractions

The number of formulas used to decompose the given improper rational functions is given below. By using the given expressions, we can quickly write the integrand as a sum of proper rational functions.

For examples,