What is #f(x) = int x+3xsqrt(x^2+1) dx# if #f(2) = 7 #?

1 Answer
Eddie
Jul 2, 2016

#\implies f(x) = x^2/2+(x^2+1)^{3/2} + 5(1- sqrt5)#

Explanation:

#f(x) = int x+3xsqrt(x^2+1) dx#

noting the pattern #D ( alpha (x^2+1)^{3/2} ) = alpha 3/2 (x^2+1)^{1/2} 2x = 3 alpha x (x^2+1)^{1/2} # so here #alpha = 1#

#\implies f(x) = x^2/2+(x^2+1)^{3/2} +C#

#7= 2+(5)^{3/2} +C implies C = 5(1- sqrt5)#

#\implies f(x) = x^2/2+(x^2+1)^{3/2} + 5(1- sqrt5)#

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