What is ₀^{π/2} a x+b x/ x+ x d x equal to ?

Correct Answer is : k(a+b)k(a+b)k(a+b)

Correct Answer is : k(a+b)k(a+b)k(a+b)

Test Index

UPSC NDA I 2026 Mathematics Paper

Question Numbers: 83-84

For the next two (02) items that follow:

\text { Let } \int_{0}^{\frac{\pi}{2}} \frac{a \sin x+b \cos x}{(a+b)(\sin x+\cos x)} d x=k

Question : 84 of 120

Marks: +1, -0

What is

\int_{0}^{\frac{\pi}{2}} \frac{a \cos x+b \sin x}{\sin x+\cos x} d x

equal to ?

Solution:

Concept:

Use the substitution

x \to \frac{\pi}{2}-x

to connect the required integral with the given one.

Explanation:

Given,

k = \int_0^{\pi/2} \frac{a\sin x+b\cos x}{(a+b)(\sin x+\cos x)}dx

Let

P = \int_0^{\pi/2} \frac{a\sin x+b\cos x}{\sin x+\cos x}dx

Since

(a+b)

is constant,

k = \frac{P}{a+b}

, so

P = k(a+b)

Let the required integral be

I = \int_0^{\pi/2} \frac{a\cos x+b\sin x}{\sin x+\cos x}dx

Put

x = \frac{\pi}{2}-t

. Then

\cos x = \sin t

and

\sin x = \cos t

Hence,

I = \int_0^{\pi/2} \frac{a\sin t+b\cos t}{\sin t+\cos t}dt = P

Therefore,

I = k(a+b)

Answer:

\displaystyle k(a+b)

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