Evaluate the indefinite integral-Calc.2 prob.

Evaluate the Indefinite integral for the following
#1
Integral of dx/x^2 sqrt of 49+x^2

#2
Integral of dx/x sqrt of x^2-25

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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Verification of second answer:
d/dx [(1/5) arctan √(x^2 - 25) + c]
= (1/5) * 1/[1 + x^2 - 25) * d/dx √(x^2 - 25)
= (1/5) * 1/(x^2 - 24) * x/√(x^2 - 25)
Yes, there seems to be some error. Let me check again.
Found the error and corrected. I had forgoten to divide by 5 inside arctan.

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