Indefinite Integration
Integration by Parts
Premium Question
Grade 12
Question:
$\int \tan^{-1}\sqrt{x} \, dx$ equals:
(1) $(x + 1)\tan^{-1}\sqrt{x} - \sqrt{x} + C$
(2) $(x + 1)\tan^{-1}\sqrt{x} + \sqrt{x} + C$
(3) $x \tan^{-1}\sqrt{x} - \sqrt{x} + C$
(4) $(x - 1)\tan^{-1}\sqrt{x} + \sqrt{x} + C$
Step-by-Step Solution
Key Concept: Apply integration by parts with $u = \tan^{-1}\sqrt{x}$ and $dv = dx$, then simplify the remaining integral using the substitution $t = \sqrt{x}$.
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Correct Answer: (1)