Q.339. Indefinite integration.

Question 339.
Evaluate the following indefinite integrals:
1) ∫ dx / [x^2 √(49 + x^2)],   2) ∫ dx / [x√(x^2 - 25)]

Answer 339.
1)
∫ dx / [x^2 √(49 + x^2)]
= ∫ dx / [x^3 √(49/x^2 + 1)]

Let 49/x^2 + 1 = t
=> - 98/x^3 dx = dt
=> Integral
= - (1/98) ∫ dt/√(t)
= - (1/98) * 2√(t) + c
= - (1/49) √(49/x^2 + 1) + c
= - √(49 + x^2) / (49x) + c.

2)
∫ dx / [x√(x^2 - 25)]
= ∫ x dx / [x^2 √(x^2 - 25)]

Let x^2 - 25 = t^2
=> x^2 = t^2 + 25 and xdx = tdt
=> Integral
= ∫ tdt / t (t^2 + 25)
= ∫ dt / (t^2 + 25
= (1/5) arctan(t/5) + c
= (1/5) arctan [(1/5)√(x^2 - 25)] + c.

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