int_2^8 frac{5^{sqrt{10-x}}}{5^{sqrt{x}}+5^{s | vedclass

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(C) Let $I = \int_2^8 \frac{5^{\sqrt{10-x}}}{5^{\sqrt{x}} + 5^{\sqrt{10-x}}} dx$  $(i)$ Using the property $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$:

Vedclass · Accepted Answer

$3$

Vedclass · Answer

(C) Let $I = \int_2^8 \frac{5^{\sqrt{10-x}}}{5^{\sqrt{x}} + 5^{\sqrt{10-x}}} dx$  $(i)$ Using the property $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$: $I = \int_2^8 \frac{5^{\sqrt{10-(2+8-x)}}}{5^{\sqrt{2+8-x}} + 5^{\sqrt{10-(2+8-x)}}} dx$ $I = \int_2^8 \frac{5^{\sqrt{10-10+x}}}{5^{\sqrt{10-x}} + 5^{\sqrt{10-10+x}}} dx$ $I = \int_2^8 \frac{5^{\sqrt{x}}}{5^{\sqrt{10-x}} + 5^{\sqrt{x}}} dx$  (ii) Adding equations $(i)$ and (ii): $2I = \int_2^8 \frac{5^{\sqrt{10-x}} + 5^{\sqrt{x}}}{5^{\sqrt{x}} + 5^{\sqrt{10-x}}} dx$ $2I = \int_2^8 1 dx$ $2I = [x]_2^8 = 8 - 2 = 6$ $I = \frac{6}{2} = 3$

$\int_2^8 \frac{5^{\sqrt{10-x}}}{5^{\sqrt{x}}+5^{\sqrt{10-x}}} d x$ is equal to

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