How do you use partial fraction decomposition to decompose the fraction to integrate x/((x+7)(x+8)(x+9))?

1 Answer
VNVDVI
Mar 31, 2018

intx/((x+7)(x+8)(x+9))dx=7/2ln|x+7|+8ln|x+8|-9/2ln|x+9|+C

Explanation:

We can decompose the integrand as follows:

x/((x+7)(x+8)(x+9))=A/(x+7)+B/(x+8)+C/(x+9)

Add up the right side:

x/((x+7)(x+8)(x+9))=(A(x+8)(x+9))/((x+7)(x+8)(x+9))+((B)(x+7)(x+9))/((x+7)(x+8)(x+9))+((C)(x+7)(x+8))/((x+7)(x+8)(x+9))

Set numerators equal:

x=A(x+8)(x+9)+B(x+7)(x+9)+C(x+7)(x+8)

We need to find A,B,C. Fortunately, it looks like we can do this by choosing particular values of x which will eliminate some terms on the right side.

x=-8:

-8=B(-1)(1), B=8

x=-7:

-7=A(-1)(2), A=7/2

x=-9:

-9=C(-2)(-1), C=-9/2

So, with the decomposed fraction, the integral becomes

7/2intdx/(x+7)+8intdx/(x+8)-9/2intdx/(x+9)

Note that all constants were factored out.

These are all simple integrals.

In general, intdx/(x+-a)=ln|x+-a|+C

Integrating yields

intx/((x+7)(x+8)(x+9))dx=7/2ln|x+7|+8ln|x+8|-9/2ln|x+9|+C

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