The value of \int_{\sqrt{e^{\pi} - e}}^{\sqrt{e^{2 \pi} - e}} \textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ }\textrm{ } \left(\right. \frac{x}{x^{2} + e} \left.\right) \textrm{ } \left|\right. \textrm{ } cos \textrm{ } \left(\right. \textrm{ } ln \textrm{ } \left(\right. x^{2} + e \left.\right) \textrm{ } \left.\right) + sin \textrm{ } \left(\right. \textrm{ } ln \textrm{ } \left(\right. x^{2} + e \left.\right) \textrm{ } \left.\right) \textrm{ } \left|\right. \textrm{ } dx is equal to

Engineering

Mathematics

Properties of Definite Integral

Question

The value of $\int_{\sqrt{e^{π} - e}}^{\sqrt{e^{2 π} - e}} (\frac{x}{x^{2} + e}) | \cos (\ln (x^{2} + e)) + \sin (\ln (x^{2} + e)) | dx$ is equal to

$2 \sqrt{2}$

$\sqrt{2} e$

$2 \sqrt{2} e$

$\sqrt{2}$

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$\int_{\sqrt{e^{π} - e}}^{\sqrt{e^{2 π} - e}} (\frac{x}{x^{2} + e}) | \cos (\ln (x^{2} + e)) + \sin (\ln (x^{2} + e)) | dx$

Put ln (x² + e) = t $\Rightarrow (\frac{x}{x^{2} + e}) dx = \frac{dt}{2}$

$I = \frac{1}{2} \int_{π}^{2 π} | cost + sint | dt = \frac{1}{2} \cdot 2 \sqrt{2} = \sqrt{2} .$