First slide

Introduction to integration

Question

$\int \sqrt{x} e^{\sqrt{x}} d x =$

Moderate

A

$2 \sqrt{x} - e^{\sqrt{x}} - 4 \sqrt{x} e^{\sqrt{x}}$
B

$(2 x - 4 \sqrt{x} + 4) e^{\sqrt{x}}$
C

$(1 - 4 \sqrt{x}) e^{\sqrt{x}}$
D

None of these

Solution

Putting $x = t^{2}, d x = 2 t d t$
$I = \int t e^{t} .2 t d t = 2 \int t^{2} e^{t} d t$ $= 2 t^{2} . e^{t} - 4 \int t e^{t} d t$

$= 2 t^{2} . e^{t} - 4 (t e^{t} - e^{t})$ $= (2 x - 4 \sqrt{x} + 4) e^{\sqrt{x}}$ .

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