What is the integral of #int (1/(x^2+x+1)) dx#? Calculus Introduction to Integration Definite and indefinite integrals 2 Answers Rhys Jul 12, 2018 #(2sqrt(3))/3 arctan( (2x+1)/sqrt(3) ) + c # Explanation: #x^2 + x+1 = (x+1/2)^2 + 3/4 # #=> int 1/((x+1/2)^2 +3/4) dx # let #3/4 u^2 = (x+1/2)^2 # #=> sqrt3 /2 u = x+1/2 # #=> sqrt(3)/2 du = dx # #=> sqrt(3)/2 int 1/(3/4 u^2 + 3/4) du # #=> (2sqrt(3))/3 int 1/(u^2+1) du # #=> (2sqrt(3))/3 arctan(u) + c # # = (2sqrt(3))/3 arctan( (2x+1)/sqrt(3) ) + "constant" # Answer link Ratnaker Mehta Jul 12, 2018 # 2/sqrt3*arctan{(2x+1)/sqrt3}+C#. Explanation: #"Prerequsite :"int1/{(x+a)^2+b^2}dx=1/b*arctan((x+a)/b)+c#. #int1/(x^2+x+1)dx#, #=int1/{(x^2+x+1/4)+3/4}dx..............."[completing square]"#, #=int1/{(x+1/2)^2+(sqrt3/2)^2}dx#, #=1/(sqrt3/2)*arctan{(x+1/2)/(sqrt3/2)}#, #=2/sqrt3*arctan{(2x+1)/sqrt3}+C#. Answer link Related questions What is the difference between definite and indefinite integrals? What is the integral of #ln(7x)#? Is f(x)=x^3 the only possible antiderivative of f(x)=3x^2? If not, why not? How do you find the integral of #x^2-6x+5# from the interval [0,3]? What is a double integral? What is an iterated integral? How do you evaluate the integral #1/(sqrt(49-x^2))# from 0 to #7sqrt(3/2)#? How do you integrate #f(x)=intsin(e^t)dt# between 4 to #x^2#? How do you determine the indefinite integrals? How do you integrate #x^2sqrt(x^(4)+5)#? See all questions in Definite and indefinite integrals Impact of this question 73875 views around the world You can reuse this answer Creative Commons License