Problem: ∫(x2+4)(x+1)x2+x+5dx
Solution:
(1) ∫(x2+4)(x+1)x2+x+5dx=∫(x2+4)(x+1)x2+4+x+1dx=I1+I2
(2) I1=∫x+11dx=ln(x+1)+C
I2=∫x2+41⋅dx
To evaluate I2 one can use the substitution
(3) x=2tan(y) then
x2+4=4(tan2(y)+1)=4(cos2(y)sin2(y)+cos2(y))=cos2(y)4;dx=cos2(y)2dy and
(4) I2=∫4/cos2(y)1⋅cos2(y)2dy=21∫dy=2y+C
Reverse substitution (3) we have y=arctan(2x).
In notation John Herschel we may write y=tan−1(2x).
(5) I2=21arctan(2x)+C
Substitute (2) and (5) in (1) we get the solution to the problem
Answer:∫(x2+4)(x+1)x2+x+5dx=ln(x+1)+21arctan(2x)+C
Comment:
(1) The solution method used here is not called "integration by parts", although the integral is divided into two parts for integration. The correct name for this method is 'fractional part integration" or 'Integration using Partial Fractions'. This method used for rational function integrals http://www.math.wsu.edu/faculty/genz/140/lessons/l506.pdf.
(2) If we use y=arctan(2x) in (3) we can immediately get
dy=arctanx′(2x)dx=2dx⋅1+(x/2)21=4+x22dx and
4+x2dx=2dy
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