Question #1aecf

1 Answer
Nov 13, 2017

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

Explanation:

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