Differentiate: f(x)=(ax^2 + bx + c) ln(x + (1+x^2)^(1/2)) + (dx + e) (1 + x^2)^(1/2). Hence integrate i) ln(x + (1 + x^2)^(1/2)), ii) (1 + x^2)^(1/2), iii) x ln(x + (1 + x^2)^(1/2)).
Differentiate equation: f'(x) = (2ax + b) ln(x + (1+x^2)^(1/2)) + ((a + 2d)x^2 + (b + c)x + (c+d)) (1 + x^2)^(-1/2).
Select correct values for constants to get:
i) x ln(x + (1+x^2)^(1/2)) - (1 + x^2)^(1/2) + C
ii) 1/2 ln(x + (1+x^2)^(1/2)) + x/2 (1 + x^2)^(1/2) + C
iii) ((x^2)/2 + 1/4) ln(x + (1+x^2)^(1/2)) - x/4 (1 + x^2)^(1/2) + C
ME
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