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Tinku Tara
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Question Number 151 by 123456 last updated on 25/Jan/15
evaluate ∫_0 ^1 ((sin x)/x)ln x dx
$$\mathrm{evaluate}\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\frac{\mathrm{sin}\:{x}}{{x}}\mathrm{ln}\:{x}\:{dx} \\ $$
Answered by prakash jain last updated on 13/Dec/14
sin x=x−(x^3 /(3!))+(x^5 /(5!))−(x^7 /(7!))+.... ((sin x)/x)=1−(x^2 /(3!))+(x^4 /(5!))−(x^6 /(7!))+.... Integrating ∫uvdx, u=ln x, v=((sin x)/x) ln x∙[x−(x^3 /(3∙3!))+...]−∫(1/x)∙[x−(x^3 /(3∙3!))+...]dx ln x∙[x−(x^3 /(3∙3!))+...]−∫[1−(x^2 /(3∙3!))+(x^4 /(5∙5!))−(x^6 /(7∙7!))...]dx ln x∙[x−(x^3 /(3∙3!))+...]−[x−(x^3 /(3^2 ∙3!))+(x^5 /(5^2 ∙5!))−(x^7 /(7^2 ∙7!))...] integrating from 0 to 1 first part is 0 result=−[1−(1/(54))+(1/(3000))−(1/(246960))+...] ≈−0.98181
$$\mathrm{sin}\:{x}={x}−\frac{{x}^{\mathrm{3}} }{\mathrm{3}!}+\frac{{x}^{\mathrm{5}} }{\mathrm{5}!}−\frac{{x}^{\mathrm{7}} }{\mathrm{7}!}+…. \\ $$$$\frac{\mathrm{sin}\:{x}}{{x}}=\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{3}!}+\frac{{x}^{\mathrm{4}} }{\mathrm{5}!}−\frac{{x}^{\mathrm{6}} }{\mathrm{7}!}+…. \\ $$$$\mathrm{Integrating}\:\int{uvdx},\:{u}=\mathrm{ln}\:{x},\:{v}=\frac{\mathrm{sin}\:{x}}{{x}} \\ $$$$\mathrm{ln}\:{x}\centerdot\left[{x}−\frac{{x}^{\mathrm{3}} }{\mathrm{3}\centerdot\mathrm{3}!}+…\right]−\int\frac{\mathrm{1}}{{x}}\centerdot\left[{x}−\frac{{x}^{\mathrm{3}} }{\mathrm{3}\centerdot\mathrm{3}!}+…\right]{dx} \\ $$$$\mathrm{ln}\:{x}\centerdot\left[{x}−\frac{{x}^{\mathrm{3}} }{\mathrm{3}\centerdot\mathrm{3}!}+…\right]−\int\left[\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\mathrm{3}\centerdot\mathrm{3}!}+\frac{{x}^{\mathrm{4}} }{\mathrm{5}\centerdot\mathrm{5}!}−\frac{{x}^{\mathrm{6}} }{\mathrm{7}\centerdot\mathrm{7}!}…\right]{dx} \\ $$$$\mathrm{ln}\:{x}\centerdot\left[{x}−\frac{{x}^{\mathrm{3}} }{\mathrm{3}\centerdot\mathrm{3}!}+…\right]−\left[{x}−\frac{{x}^{\mathrm{3}} }{\mathrm{3}^{\mathrm{2}} \centerdot\mathrm{3}!}+\frac{{x}^{\mathrm{5}} }{\mathrm{5}^{\mathrm{2}} \centerdot\mathrm{5}!}−\frac{{x}^{\mathrm{7}} }{\mathrm{7}^{\mathrm{2}} \centerdot\mathrm{7}!}…\right] \\ $$$$\mathrm{integrating}\:\mathrm{from}\:\mathrm{0}\:\mathrm{to}\:\mathrm{1}\:\mathrm{first}\:\mathrm{part}\:\mathrm{is}\:\mathrm{0} \\ $$$$\mathrm{result}=−\left[\mathrm{1}−\frac{\mathrm{1}}{\mathrm{54}}+\frac{\mathrm{1}}{\mathrm{3000}}−\frac{\mathrm{1}}{\mathrm{246960}}+…\right] \\ $$$$\approx−\mathrm{0}.\mathrm{98181} \\ $$

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