∫040142x2x+24014−xdx=

I=∫040142x2x+24014−xdxI=∫0401424014−x24014+2xdx[∴∫0af(x)dx=∫0af(a−x)dx]I+I=∫040142x+24014−x2x+24014−xdx2I=∫040141 dxx04014=4014I=2007

∫040142x2x+24014−xdx=

I=∫040142x2x+24014−xdxI=∫0401424014−x24014+2xdx[∴∫0af(x)dx=∫0af(a−x)dx]I+I=∫040142x+24014−x2x+24014−xdx2I=∫040141 dxx04014=4014I=2007

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$\int_{0}^{4014} \frac{2^{x}}{2^{x} + 2^{4014 - x}} d x =$

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$2^{2007}$

$2^{4014}$

4014

2007

answer is D.

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Detailed Solution

$\begin{array}{l} I = \int_{0}^{4014} \frac{2^{x}}{2^{x} + 2^{4014 - x}} d x \\ I = \int_{0}^{4014} \frac{2^{4014 - x}}{2^{4014} + 2^{x}} d x \\ [∴ \int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x] \\ I + I = \int_{0}^{4014} \frac{2^{x} + 2^{4014 - x}}{2^{x} + 2^{4014 - x}} d x \\ 2 I = \int_{0}^{4014} 1 d x \\ {(x)}_{0}^{4014} = 4014 \\ I = 2007 \end{array}$