How do you evaluate the definite integral #int (x^3+2x)/(x^2+1)# from #[0, 2]#?

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Answer 1 · 2018-05-07T16:52:01

The answer is #=2.8047#

First, calculate the indefinite integral by substitution

Let #u=x^2+1#, #=>#, #du=#2xdx#

Therefore,

The indefinite integral is

#I=int((x^3+2x)dx)/(x^2+1)=int((x^2+2)xdx)/(x^2+1)#

#=1/2int((u+1)du)/u#

#=1/2int(1+1/u)du#

#=1/2(u+lnu)#

#=1/2(x^2+1)+1/2ln(x^2+1)+C#

The definite integral is

#int_0^2((x^3+2x)dx)/(x^2+1)=[1/2(x^2+1)+1/2ln(x^2+1)]_0^2#

#=(5/2+1/2ln(5))-(1/2+0)#

#=2+1/2ln(5)#

#=2.8047#