Indefinite Integration
Indefinite Integration
nta_pyq_2025_jan
Grade 12
Question:
If \int e ( x x sin -1 x + sin -1 x + x ) dx = g(x) + C , where C is the constant of integration, then g ( 1 ) equals : 3/2 1-x 2 2 \sqrt1-x2 (1-x ) 2
\pi 4 \sqrt e 3
\pi 6 \sqrt e 3
\pi 4 \sqrt e 2
\pi 6 \sqrt e 2
Step-by-Step Solution
Key Concept: Apply the core result for integration by substitution and identities and simplify using the given constraints.
x⋅2x \sqrt1 - x 2 + -1 ⎛ 1 ⋅ ⎞ d x ⋅ sin x 2\sqrt1-x 2 -1 (2) ( ) - sin x ⋅ ⎜ ⎜ 2 ⎟ ⎟ dx \sqrt1 - x 2 1 - x ⎝ ⎠ x 1 = ⋅ \sqrt1 - x 2 \sqrt1 - x 2 -1 sin x x = + 3/2 2 2 1 - x (1 - x ) x ′ Hence, I = \int e (f (x) + f (x)) dx x = e ⋅ f (x) + C -1 x ⋅ sin x x I = e ⋅ + C = g(x) + C \sqrt1 - x 2 x -1 xe sin x \pi e \Rightarrow g(x) = and g(1/2) = \sqrt \sqrt1 - x 2 6 3
Correct Answer: 2