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  • Prove the following (Exercises 34 to 39) Integral 0 to 1 x e ^x dx = 1

Prove the following (Exercises 34 to 39)

    Q35.    \int_0^1 xe^xdx =1

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Let\ I=\int xe^{x}dx

Integrating I by parts

\\I=x\int e^{x}dx-\int ( (\frac{\mathrm{d} (x)}{\mathrm{d} x})\int e^{x}dx)dx\\ I=xe^{x}-\int e^{x}dx\\ I=xe^{x}-e^{x}+c

Applying Limits from 0 to 1

\\\int_{0}^{1}xe^{x}dx=[xe^{x}-e^{x}+c]_{0}^{1}\\ I=[e-e+c]-[0-1+c]\\ I=1

Hence proved I = 1

 

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Sayak

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