Use partial fractions to find the indefinite integral. ∫x2+12x+12x3−4xdx

piarepm

piarepm

Answered

2022-01-07

Use partial fractions to find the indefinite integral.
x2+12x+12x34xdx

Answer & Explanation

Bob Huerta

Bob Huerta

Expert

2022-01-08Added 41 answers

Step 1
Given: I=x2+12x+12x34xdx
For evaluating given integral, first we simplify given expression then integrate it
Step 2
So,
I=x2+12x+12x34xdx
=x3+12x+12x(x24)dx
=x3+12x+12x(x222)dx   (a2b2=(a+b)(ab))
=x2+12x+12x(x+2)(x2)dx
=(5x23x1x+2)dx   (dxx+a=ln|x+a|+c)
=5ln|x2|3ln|x|ln|x+2|+c
Hence, given integral can be find as above.
poleglit3

poleglit3

Expert

2022-01-09Added 32 answers

x2+12x+12x34xdx
x2+12x+12x34x=Ax+Bx2+Cx+2
x2+12x+12=A(x24)+B×(x+2)+C×(x2)
put x=0
12=A(4)+B×0+C×0
A=3
put x=2
4+24+12=B2(2+2)
40=B×8
B=5
put x=2,(2)2+12(2)+12=A((2)24)+B(2)(22)+C(2)(22)
424+12=8C
8=8CC=1
x2+12x+12x34x=5x21x+23x
x2+12x+12x34xdx=5dxx2dxx+23dxx
=5ln(x2)ln(x+2)3lnx+C
=ln(x2)5x3(x+2)+C

karton

karton

Expert

2022-01-11Added 439 answers

Given:
x2+12x+12x34xdx3x+5x21x+2dx3xdx+5x2dx1x+2dx3ln(|x|)+5ln(|x2|)ln(|x+2|)Answer:3ln(|x|)+5ln(|x2|)ln(|x+2|)+C

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