evaluate-0-1-sin-x-x-ln-x-dx-

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Question Number 151 by 123456 last updated on 25/Jan/15

$$\mathrm{evaluate}\underset{0}{\stackrel{1}{\int }}\frac{\sin x}{x}\ln xdx$$

Answered by prakash jain last updated on 13/Dec/14

$$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\dots .$$$$\frac{\sin x}{x}=1-\frac{x^2}{3!}+\frac{x^4}{5!}-\frac{x^6}{7!}+\dots .$$$$\mathrm{Integrating}\int uvdx,u=\ln x,v=\frac{\sin x}{x}$$$$\ln x\cdot [x-\frac{x^3}{3\cdot 3!}+\dots ]-\int \frac{1}{x}\cdot [x-\frac{x^3}{3\cdot 3!}+\dots ]dx$$$$\ln x\cdot [x-\frac{x^3}{3\cdot 3!}+\dots ]-\int [1-\frac{x^2}{3\cdot 3!}+\frac{x^4}{5\cdot 5!}-\frac{x^6}{7\cdot 7!}\dots ]dx$$$$\ln x\cdot [x-\frac{x^3}{3\cdot 3!}+\dots ]-[x-\frac{x^3}{3^2\cdot 3!}+\frac{x^5}{5^2\cdot 5!}-\frac{x^7}{7^2\cdot 7!}\dots ]$$$$\mathrm{integrating}\mathrm{from}0\mathrm{to}1\mathrm{first}\mathrm{part}\mathrm{is}0$$$$\mathrm{result}=-[1-\frac{1}{54}+\frac{1}{3000}-\frac{1}{246960}+\dots ]$$$$\approx -0.98181$$

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