Find integrals of given function- -e-x-x dx -

=∫ e^√(x)√(x)dx √(x)=t #160; #160; x=t^2 dx2√(x)=dt #160; #160; #160; √(x)dx2x=dt√(x)=2xdt=2t^2dt =∫ 2t^2e^tdt ...

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Byju's Answer

Standard XII

Mathematics

Integration by Substitution

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Question

Find integrals of given function; $\int e^{\sqrt{x}} \sqrt{x} d x =$

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Solution

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

$\sqrt{x} = t$ $x = t^{2}$

$\frac{d x}{2 \sqrt{x}} = d t$ $\frac{\sqrt{x} d x}{2 x} = d t \sqrt{x} = 2 x d t = 2 t^{2} d t$
$= \int 2 t^{2} e^{t} d t$

$= 2 [t^{2} e^{t} - \int 2 t e^{t} d t]$
$= 2 [t^{2} e^{t} - 2 t e^{t} + 2 \int e^{t} d t]$

$= 2 [t^{2} e^{t} - 2 t e^{t} + 2 e^{t}] + C$

put $t = \sqrt{x}$

$= 2 x e^{\sqrt{x}} - 4 \sqrt{x} e^{\sqrt{x}} + 4 e^{\sqrt{x}} + C$ .

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Find integrals of given function; ∫e√x√xdx=

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Find integrals of given function; $\int e^{\sqrt{x}} \sqrt{x} d x =$