Question:
$\int\limits^{\pi/2}_{0} \frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}} dx $ equals
$\int\limits^{\pi/2}_{0} \frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}} dx $ equals
Updated On: Mar 18, 2024
- 2
- $\pi$
- $\pi / 4$
- $\pi / 2$
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The Correct Option is C
Solution and Explanation
$I = \int\limits^{\pi/2}_{0} \frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}} dx $
$ I = \int\limits^{\pi/2}_{0} \frac{2^{\sin\left(\pi/2-x\right)}}{2^{\sin\left(\pi/2-x\right)} + 2^{\cos\left(\pi/2-x\right)}}dx $
$ = \int \frac{2^{\cos x}}{2^{\cos x} + 2^{\sin x}} dx $
$ \Rightarrow 21 = \int\limits^{\pi/2}_{0} dx = \frac{\pi}{2}$
$ \Rightarrow 1 = \frac{\pi}{4} $
$ I = \int\limits^{\pi/2}_{0} \frac{2^{\sin\left(\pi/2-x\right)}}{2^{\sin\left(\pi/2-x\right)} + 2^{\cos\left(\pi/2-x\right)}}dx $
$ = \int \frac{2^{\cos x}}{2^{\cos x} + 2^{\sin x}} dx $
$ \Rightarrow 21 = \int\limits^{\pi/2}_{0} dx = \frac{\pi}{2}$
$ \Rightarrow 1 = \frac{\pi}{4} $
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Concepts Used:
Integrals of Some Particular Functions
There are many important integration formulas which are applied to integrate many other standard integrals. In this article, we will take a look at the integrals of these particular functions and see how they are used in several other standard integrals.
Integrals of Some Particular Functions:
- ∫1/(x2 – a2) dx = (1/2a) log|(x – a)/(x + a)| + C
- ∫1/(a2 – x2) dx = (1/2a) log|(a + x)/(a – x)| + C
- ∫1/(x2 + a2) dx = (1/a) tan-1(x/a) + C
- ∫1/√(x2 – a2) dx = log|x + √(x2 – a2)| + C
- ∫1/√(a2 – x2) dx = sin-1(x/a) + C
- ∫1/√(x2 + a2) dx = log|x + √(x2 + a2)| + C
These are tabulated below along with the meaning of each part.
