What is equal to

Question

Question

Direction : Consider the following for the items that follow :

Let , where f(x) = sin x and g(x) = sin x + cos x + 1

This question was previously asked in

NDA-II 2024 (Maths) Official Paper (Held On: 01 Sept, 2024)

Answer 1

Concept:

Integration using Substitution:

This method involves replacing a complex expression with a single variable to simplify integration.
Useful for trigonometric integrals where identities and substitutions simplify the expression.
Here, the identity used is:
The integral is solved using limits transformation and properties of logarithmic integrals.

Calculation:

Given:

We evaluate

⇒ Apply Weierstrass substitution:

⇒ Use identity:

⇒ Simplify denominator:

⇒

Let

⇒ Limits change:

⇒

∴ The final answer is

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