Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #77

Evaluate \(\int \operatorname{asin}{\left(\sqrt{\frac{x}{a + x}} \right)}\,dx\). Work for a>0 and x>0.

✅ Step-by-Step Solution

Final Answer: \(- \sqrt{a x} + \left(a + x\right) \operatorname{atan}{\left(\sqrt{\frac{x}{a}} \right)}\) + C

Assume a>0,x>0 and let θ=arctan√(x/a)∈(0,π/2). Then x=a tan²θ and arcsin√[x/(a+x)]=θ.
The integral is 2a∫θ tanθ sec²θ dθ. By parts it equals aθtan²θ−a∫tan²θ dθ.
Using tan²θ=sec²θ−1 gives aθtan²θ−a tanθ+aθ; substitute x and θ.
Therefore the required integral is \(- \sqrt{a x} + \left(a + x\right) \operatorname{atan}{\left(\sqrt{\frac{x}{a}} \right)}\) + C.

[0.5 marks] Assume a>0,x>0 and let θ=arctan√(x/a)∈(0,π/2). Then x=a tan²θ and arcsin√[x/(a+x)]=θ.
[1 mark] The integral is 2a∫θ tanθ sec²θ dθ. By parts it equals aθtan²θ−a∫tan²θ dθ.
[1 mark] Using tan²θ=sec²θ−1 gives aθtan²θ−a tanθ+aθ; substitute x and θ.
[0.5 marks] Therefore the required integral is \(- \sqrt{a x} + \left(a + x\right) \operatorname{atan}{\left(\sqrt{\frac{x}{a}} \right)}\) + C.

Practise interactive questions, flashcards, visual mind maps, and timed board tests for Integrals.