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 10913 by Saham last updated on 01/Mar/17

∫_( 1) ^( 3) x^x dx

$$\int_{\:\mathrm{1}} ^{\:\mathrm{3}} \:\mathrm{x}^{\mathrm{x}} \:\:\mathrm{dx} \\ $$

Answered by FilupS last updated on 02/Mar/17

I=∫_1 ^( 3) x^x dx I=∫_1 ^( 3) e^(xln(x)) dx e^t =Σ_(n=0) ^∞ (t^n /(n!)) ∴ x^x =e^(xln(x)) =Σ_(n=0) ^∞ ((x^n ln^n (x))/(n!)) I=∫_1 ^( 3) Σ_(n=0) ^∞ ((x^n ln^n (x))/(n!))dx I=Σ_(n=0) ^∞ ∫_1 ^( 3) ((x^n ln^n (x))/(n!))dx I=Σ_(n=0) ^∞ (1/(n!))∫_1 ^( 3) x^n ln^n (x)dx

$${I}=\int_{\mathrm{1}} ^{\:\mathrm{3}} {x}^{{x}} {dx} \\ $$$${I}=\int_{\mathrm{1}} ^{\:\mathrm{3}} {e}^{{x}\mathrm{ln}\left({x}\right)} {dx} \\ $$$${e}^{{t}} =\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{t}^{{n}} }{{n}!} \\ $$$$\therefore\:{x}^{{x}} ={e}^{{x}\mathrm{ln}\left({x}\right)} =\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{x}^{{n}} \mathrm{ln}^{{n}} \left({x}\right)}{{n}!} \\ $$$${I}=\int_{\mathrm{1}} ^{\:\mathrm{3}} \underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{x}^{{n}} \mathrm{ln}^{{n}} \left({x}\right)}{{n}!}{dx} \\ $$$${I}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\int_{\mathrm{1}} ^{\:\mathrm{3}} \frac{{x}^{{n}} \mathrm{ln}^{{n}} \left({x}\right)}{{n}!}{dx} \\ $$$${I}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}!}\int_{\mathrm{1}} ^{\:\mathrm{3}} {x}^{{n}} \mathrm{ln}^{{n}} \left({x}\right){dx} \\ $$

Commented by Saham last updated on 02/Mar/17

God bless you sir.

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com