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Question Number 65961 by mmkkmm000m last updated on 06/Aug/19

$$\mathrm{Let}\frac{d}{dx}\left(F\left(x\right)\right)=\frac{e^{\sin x}}{x},x>0.$$$$\mathrm{If}\underset{1}{\stackrel{4}{\int }}\frac{2e^{\sin x^2}}{x}dx=F\left(k\right)-F\left(1\right),\mathrm{then}\mathrm{one}$$$$\mathrm{of}\mathrm{the}\mathrm{possible}\mathrm{values}\mathrm{of}k\mathrm{is}$$

Answered by mr W last updated on 06/Aug/19

$$\underset{1}{\stackrel{4}{\int }}\frac{2e^{\sin x^2}}{x}dx$$$$=\underset{1}{\stackrel{4}{\int }}\frac{2x{e}^{\sin x^2}}{x^2}dx$$$$=\underset{1}{\stackrel{4}{\int }}\frac{e^{\sin x^2}}{x^2}d\left(x^2\right)$$$$=\underset{1}{\stackrel{16}{\int }}\frac{e^{\sin t}}{t}dtwitht=x^2$$$$={\left[F\left(t\right)\right]}_1^{16}$$$$=F\left(16\right)-F\left(1\right)$$$$=F\left(k\right)-F\left(1\right)$$$$\Rightarrow k=16$$

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