How do you find the integral intx/(sqrt(x^2+x+1))dx ?
1 Answer
May 22, 2018
Use the substitution
Explanation:
Let
I=intx/sqrt(x^2+x+1)dx
Complete the square in the denominator:
I=int(2x)/sqrt((2x+1)^2+3)dx
Apply the substitution
I=int(sqrt3tantheta-1)/(sqrt3sectheta)(sqrt3/2sec^2thetad theta)
Simplify:
I=1/2int(sqrt3secthetatantheta-sectheta)d theta
Integrate term by term:
I=1/2{sqrt3sectheta-ln|sectheta+tantheta|}+C
Reverse the substitution:
I=1/2sqrt((2x+1)^2+3)-1/2ln|2x+1+sqrt((2x+1)^2+3)|+C