Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 122120 by sina1377 last updated on 14/Nov/20

∫_0 ^3 (1/( (√y))).e^y dy

$$\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}\frac{\mathrm{1}}{\:\sqrt{{y}}}.{e}^{{y}} {dy} \\ $$

Answered by Bird last updated on 14/Nov/20

I =∫_0 ^3 (e^y /( (√y)))dy ⇒I =_((√y)=x) ∫_0 ^(√3) (e^x^2 /x)(2xdx) =2∫_0 ^(√3) e^x^2 dx =2∫_0 ^(√3) Σ_(n=0) ^∞ (x^(2n) /(n!))dx =2Σ_(n=0) ^∞ (1/(n!))∫_0 ^(√3) x^(2n) dx =2Σ_(n=0) ^∞ (1/(n!))[(1/(2n+1)) x^(2n+1) ]_0 ^(√3) =2 Σ_(n=0) ^∞ (1/((2n+1)n!))((√3))^(2n+1) =2(√3)Σ_(n=0) ^∞ (3^n /((2n+1)n!))

$${I}\:=\int_{\mathrm{0}} ^{\mathrm{3}} \:\frac{{e}^{{y}} }{\:\sqrt{{y}}}{dy}\:\:\Rightarrow{I}\:=_{\sqrt{{y}}={x}} \:\:\:\int_{\mathrm{0}} ^{\sqrt{\mathrm{3}}} \:\frac{{e}^{{x}^{\mathrm{2}} } }{{x}}\left(\mathrm{2}{xdx}\right) \\ $$$$=\mathrm{2}\int_{\mathrm{0}} ^{\sqrt{\mathrm{3}}} \:{e}^{{x}^{\mathrm{2}} } {dx}\:=\mathrm{2}\int_{\mathrm{0}} ^{\sqrt{\mathrm{3}}} \sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{x}^{\mathrm{2}{n}} }{{n}!}{dx} \\ $$$$=\mathrm{2}\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{{n}!}\int_{\mathrm{0}} ^{\sqrt{\mathrm{3}}} \:{x}^{\mathrm{2}{n}} {dx} \\ $$$$=\mathrm{2}\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{{n}!}\left[\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{1}}\:{x}^{\mathrm{2}{n}+\mathrm{1}} \right]_{\mathrm{0}} ^{\sqrt{\mathrm{3}}} \\ $$$$=\mathrm{2}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{\left(\mathrm{2}{n}+\mathrm{1}\right){n}!}\left(\sqrt{\mathrm{3}}\right)^{\mathrm{2}{n}+\mathrm{1}} \\ $$$$=\mathrm{2}\sqrt{\mathrm{3}}\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{3}^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right){n}!} \\ $$

Answered by Dwaipayan Shikari last updated on 14/Nov/20

∫_0 ^3 Σ_(n=0) ^∞ (y^(n−(1/2)) /(n!)) Σ_(n=0) ^∞ ∫_0 ^3 (y^(n−(1/2)) /(n!))=2Σ_(n=0) ^∞ (3^(n+(1/2)) /(n!(2n+1))) 2(√3)Σ_(n=0) ^∞ (3^n /(n!(2n+1)))

$$\int_{\mathrm{0}} ^{\mathrm{3}} \underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{y}^{{n}−\frac{\mathrm{1}}{\mathrm{2}}} }{{n}!} \\ $$$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\int_{\mathrm{0}} ^{\mathrm{3}} \frac{{y}^{{n}−\frac{\mathrm{1}}{\mathrm{2}}} }{{n}!}=\mathrm{2}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{3}^{{n}+\frac{\mathrm{1}}{\mathrm{2}}} }{{n}!\left(\mathrm{2}{n}+\mathrm{1}\right)} \\ $$$$\mathrm{2}\sqrt{\mathrm{3}}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{3}^{{n}} }{{n}!\left(\mathrm{2}{n}+\mathrm{1}\right)} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com