∫_0^1 xe^x dx= 1 Using integration by parts nbsp; = [ x ∫ e^x ∫ ( ddx(x)∫ e^x dx ) dx ]^1_0 = [ xe^x ∫ (1.e^x) dx ]^1_0 = [xe^x e^x]^1_0 =[(1e^1 e^1) ...

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Standard XII

Mathematics

Degree of a Differential Equations

Prove ∫01 xex...

Question

Prove $\int_{0}^{1} x e^{x} d x = 1$

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Solution

$\int_{0}^{1} x e^{x} d x = 1$
Using integration by parts
$= {[x \int e^{x} - \int (\frac{d}{d x} (x) \int e^{x} d x) d x]}_{0}^{x}$
$= {[x e^{x} - \int (1. e^{x}) d x]}_{0}^{x}$
$= [x e^{x} - e^{x}]_{0}^{1}$
$= [(1 e^{1} - e^{1}) - (0 - e^{0})]$
$= 1$
Hence proved.

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Degree of a Differential Equations

Standard XII Mathematics

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Prove ∫10xexdx=1

∫10xexdx=1 Using integration by parts =[x∫ex−∫(ddx(x)∫exdx)dx]10 =[xex−∫(1.ex)dx]10 =[xex−ex]10 =[(1e1−e1)−(0−e0)] =1 Hence proved.

Prove $\int_{0}^{1} x e^{x} d x = 1$

$\int_{0}^{1} x e^{x} d x = 1$
Using integration by parts
$= {[x \int e^{x} - \int (\frac{d}{d x} (x) \int e^{x} d x) d x]}_{0}^{x}$
$= {[x e^{x} - \int (1. e^{x}) d x]}_{0}^{x}$
$= [x e^{x} - e^{x}]_{0}^{1}$
$= [(1 e^{1} - e^{1}) - (0 - e^{0})]$
$= 1$
Hence proved.