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Evaluate the $\int\limits_0^1\int\limits_0^xe^{x+y}dydx$

written 9.5 years ago by teamques10 ★ 70k

•   modified 3.9 years ago

engineering mathematics

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written 9.5 years ago by teamques10 ★ 70k

Integrating first w.r.t. y

$$I=\int\limits_0^1e^x[e^y]^x_0dx$$

$I=\int\limits_0^1e^x[e^x-1]dx\\ =\int\limits_0^1e^{2x}-e^xdx\\ I=[e^{2x}-e^x]^1_0\\ =\Bigg[\dfrac {e^2}2-e^1\Bigg]-\Bigg[\dfrac 12-1\Bigg]\\ =\dfrac {e^2}2-e^1+\dfrac 12\\ =\dfrac 12[e^2-2e^1+1]\\ =\dfrac 12(e-1)^2$

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