Indefinite Integration
Inverse Trigonometric Integrals
Grade 12
Question:
<p>For \(a > 0\), if \(I = \int \frac{x dx}{a^3 - x^3} = A \sin^{-1}\left( \frac{x^{3/2}}{B} \right) + C\), where \(C\) is any arbitrary constant, then:</p>
<p>(a) \(A = \frac{2}{3}\)</p>
<p>(b) \(B = a^{3/2}\)</p>
<p>(c) \(A = \frac{1}{3}\)</p>
<p>(d) \(B = a^{1/2}\)</p>
Step-by-Step Solution
Key Concept: Apply appropriate substitution for irrational algebraic integrals.
<p>This is identical to question 3. Substitution $u = x^{3/2}$ transforms the integral to standard form with $A = \frac{2}{3}$ and $B = a^{3/2}$.</p>
Correct Answer: a, b