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Answer 1 · 2017-11-13T06:35:22

#-7/(2(5+x^2)^2) + C#

#int(14x)/(5+x^2)^3dx#

#=7int(2x)/(5+x^2)^3dx#

#=-7/2(5+x^2)^-2 + C# by reverse chain rule

#=-7/(2(5+x^2)^2) + C#

The reverse chain rule can be more easily seen if we use a substitution.

Let #u=5+x^2#

#(du)/dx=2x#

#:.du=2xdx#

#:.7int(2x)/(5+x^2)^3dx=7int(du)/u^3#

#=7[-u^-2/2]+C#

#=-7/(2(5+x^2)^2) + C#

Checking result:

#d/dx(-7/(2(5+x^2)^2) + C)#

#=7(5+x^2)^-3*2x# by chain rule

#=7/(5+x^2)^3*2x#

#=(14x)/(5+x^2)^3#