integration

(1) 5x^2 + 50 x + 118 = 5(x + 5)^2 - 7.
So firstly let x + 5 = u, the integral becomes
(u - 5) sqrt (5u^2 - 7) = u sqrt (5u^2 - 7) - 5 sqrt (5u^2 - 7).
The first part can be integrated simply by substitution method.
For the second part, it belongs to the standard form ∫ sqrt ( x^2 - a^2) dx that use the trig. substitution x = a sec t. For this question, it is u = sqrt(7/5) sec t or x + 5 = sqrt(7/5) sec t.
(2)
- 7x^2 + 70x - 172 = - 7(x - 5)^2 + 3
So firstly let x - 5 = u, the integral becomes
(u + 5)/sqrt [3 - 7u^2] = u/sqrt ( 3 - 7u^2) + 5/sqrt(3 - 7u^2)
The first part can be integrated by substitution method.
For the second part, it belongs to the form ∫ 5/sqrt(a^2 - x^2) dx that use the trig. substitution x = a sin t. For this question, it is u = sqrt(3/7) sin t or x - 5 = sqrt(3/7) sin t.