∫0^n / 2 (√(ain x))/(√(sinx)+√(cosx)) d x in equal to (a) 0 (b) 1 (c) (π)/(4) ( d ) (5)/(2).

Name: Problem 2, Chapter 6: Higher Engineering Mathematics
Uploaded: 2022-04-03T19:07:16-08:00
Duration: 1 min 57 s
Channel: Khaled Yasein
Description: ∫0^n / 2 (√(ain x))/(√(sinx)+√(cosx)) d x in equal to (a) 0 (b) 1 (c) (π)/(4) ( d ) (5)/(2).

step 1 In this problem we have i which is equal to the integral from 0 to pi over 2 of square root of s

Question

$\int_{0}^{n / 2} \frac{\sqrt{\operatorname{ain} x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x$ in equal to (a) 0 (b) $\underline{1}$ (c) $\frac{\pi}{4}$ $($ d $) \frac{5}{2}$.

   $\int_{0}^{n / 2} \frac{\sqrt{\operatorname{ain} x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x$ in equal to
(a) 0
(b) $\underline{1}$
(c) $\frac{\pi}{4}$
$($ d $) \frac{5}{2}$.

Higher Engineering Mathematics

Grewal B.S. 42nd Edition

Chapter 6, Problem 2

Instant Answer

Solved by Expert Khaled Yasein

04/03/2022

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We need to analyze the integrand and the limits of integration. Show more…

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$\int_{0}^{n / 2} \frac{\sqrt{\operatorname{ain} x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x$ in equal to (a) 0 (b) $\underline{1}$ (c) $\frac{\pi}{4}$ $($ d $) \frac{5}{2}$.

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Key Concepts

Trigonometric Function Symmetry

Trigonometric functions, such as sine and cosine, exhibit symmetry properties, especially over intervals like [0, ?/2]. These complementary relationships allow for the transformation of integrals by interchanging sine and cosine, which can simplify evaluation by revealing constant sums in paired integrals.

Substitution Method in Integration

The substitution method is a technique for simplifying integrals by changing variables. In trigonometric integrals, a common substitution is using the complementary angle (for example, replacing x with ?/2 - x) to exploit symmetry, reduce complexity, and combine integrals to achieve a form that is easier to integrate.

Definite Integrals

A definite integral represents the accumulation of quantities, typically areas under a curve, over a specific interval. It computes the net result of applying a function over a given range and is fundamental in calculus for quantifying continuous sums.

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