How do you integrate?
1 Answer
Mar 11, 2018
Use the substitution
Explanation:
Let
#I=int(x+9)/(x^2+8x+17)^2dx#
Complete the square in the square root:
#I=int(x+9)/((x+4)^2+1)^2dx#
Apply the substitution
#I=int(tantheta+5)/(sec^4theta)(sec^2thetad theta)#
Simplify:
#I=int(sinthetacostheta+5cos^2theta)d theta#
Apply the double-angle trigonometric identities:
#I=1/2int(5+sin2theta+5cos2theta)d theta#
Integrate directly:
#I=1/2(5theta-1/2cos2theta+5/2sin2theta)+C#
Rearrange:
#I=5/2theta+(5tantheta-1)/(2sec^2theta)+C#
Reverse the substitution:
#I=5/2tan^(-1)(x+4)+1/2(5x+19)/(x^2+8x+17)+C#