Solved: "Evaluate the following integral. ∫01∫0x2ex2dydx"

sibuzwaW

Answered question

2021-11-01

Evaluate the following integral.

\int_{0}^{1} \int_{0}^{x} 2 e^{x^{2}} d y d x

hosentak

Skilled2021-11-02Added 100 answers

Given,

\int_{0}^{1} \int_{0}^{x} 2 e^{x^{2}} d y d x

On integrating, we get

\int_{0}^{1} \int_{0}^{x} 2 e^{x^{2}} d y d x = \int_{0}^{1} 2 e^{x^{2}} \cdot {[y]}_{0}^{x} d x

= \int_{0}^{1} 2 e^{x^{2}} \cdot [x - 0] d x

= \int_{0}^{1} 2 x e^{x^{2}} d x

Step 2
Now put

x^{2} = t

\Rightarrow 2 x = \frac{d t}{d x}

\Rightarrow 2 x d x = d t

When x=0, t=0
& when x=1, t=1
Therefore,

\int_{0}^{1} 2 x e^{x^{2}} d x = \int_{0}^{1} e^{t} d t

= {[e^{t}]}_{0}^{1}

= e^{1} - e^{0}

=e-1

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