The value of int_{0}^{pi /2} frac{2^{sin x}}{ | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $I = \int_{0}^{\pi /2} \frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}} dx$ ....$(i)$Using the property $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x)

Vedclass · Accepted Answer

$\frac{\pi}{4}$

Vedclass · Answer

(A) Let $I = \int_{0}^{\pi /2} \frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}} dx$ ....$(i)$Using the property $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$,we get:$I = \int_{0}^{\pi /2} \frac{2^{\sin(\pi/2 - x)}}{2^{\sin(\pi/2 - x)} + 2^{\cos(\pi/2 - x)}} dx$Since $\sin(\pi/2 - x) = \cos x$ and $\cos(\pi/2 - x) = \sin x$,we have:$I = \int_{0}^{\pi /2} \frac{2^{\cos x}}{2^{\cos x} + 2^{\sin x}} dx$ ....$(ii)$Adding equations $(i)$ and $(ii)$:$2I = \int_{0}^{\pi /2} \left( \frac{2^{\sin x} + 2^{\cos x}}{2^{\sin x} + 2^{\cos x}} \right) dx$$2I = \int_{0}^{\pi /2} 1 dx$$2I = [x]_{0}^{\pi /2} = \frac{\pi}{2}$$I = \frac{\pi}{4}$.

The value of $\int_{0}^{\pi /2} \frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}} dx$ is

Similar Questions

The value of the integral $\int_{-\pi}^{\pi} (\cos px - \sin qx)^2 dx$,where $p$ and $q$ are integers,is equal to:

If $[.]$ denotes the greatest integer function,then $\int_0^{1000} e^{x-[x]} dx=$

$\left[ {\sum\limits_{n = 1}^{10} {\int_{ - 2n - 1}^{2n} {{{\sin }^{27}}x\,dx} } } \right] + \left[ {\sum\limits_{n = 1}^{10} {\int_{2n}^{2n + 1} {{{\sin }^{27}}x\,dx} } } \right]$ equals

$\int\limits_0^\infty {\frac{{{x^3}}}{{1 + x + 2{x^2} + 2{x^3} + {x^4} + {x^5}}}} dx$

The value of $\int_{0}^{\pi /2} \frac{\sin^{2/3} x}{\sin^{2/3} x + \cos^{2/3} x} dx$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module