How do you integrate?

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1 Answer
Mar 11, 2018

Use the substitution #x+4=tantheta#.

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 #x+4=tantheta#:

#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#