How do you find the antiderivative?

Question

#dx(2+(x^2))/(1+(x^2))#

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Answer 1 · 2018-05-04T08:38:02

#int (2+x^2)/(1+x^2) dx = arctanx +x+C#

Find the primitive by solving the indefinite integral:

#int (2+x^2)/(1+x^2) dx#

Split the numerator:

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

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

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

Using the linearity if the integral:

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

which are well known integrals:

#int (2+x^2)/(1+x^2) dx = arctanx +x+C#