Class 12 Integrals · Long Answer3 Marksindefinite_integration
Class 12 Maths · Integrals · Long Answer #82
Evaluate \(\int \frac{\sqrt{x}}{\sqrt{a^{3} - x^{3}}}\,dx\). Work for a>0 and 0<x<a.
✅ Step-by-Step Solution
Final Answer: \(\frac{2 \operatorname{asin}{\left(\left(\frac{x}{a}\right)^{\frac{3}{2}} \right)}}{3}\) + C
- Assume a>0 and 0<x<a. Let u=x^(3/2), du=(3/2)√x dx.
- The integral becomes (2/3)∫du/√[(a^(3/2))²−u²].
- Apply the arcsine formula to obtain (2/3)arcsin[u/a^(3/2)] and substitute u=x^(3/2).
- Therefore the required integral is \(\frac{2 \operatorname{asin}{\left(\left(\frac{x}{a}\right)^{\frac{3}{2}} \right)}}{3}\) + C.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Assume a>0 and 0<x<a. Let u=x^(3/2), du=(3/2)√x dx.
- [1 mark] The integral becomes (2/3)∫du/√[(a^(3/2))²−u²].
- [1 mark] Apply the arcsine formula to obtain (2/3)arcsin[u/a^(3/2)] and substitute u=x^(3/2).
- [0.5 marks] Therefore the required integral is \(\frac{2 \operatorname{asin}{\left(\left(\frac{x}{a}\right)^{\frac{3}{2}} \right)}}{3}\) + C.
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