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[Solved] Integral

 
Integral
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June 15, 2021 2:51 pm
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(@armor)
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Find \int \frac{x^6-3x^5+2x^4+x^3-x^2+4x-7}{(x-2)^3}dx.

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June 15, 2021 2:51 pm
Topic starter
Staff 
(@armor)
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317 Posts
156 153 0

First we write the numerator as 

\small x^5(x-2)-x^4(x-2)+x^2(x-2)+x(x-2)+6(x-2)+5

\small =(x-2)[x^4(x-2)+x^3(x-2)+2x^2(x-2)+5x(x-2)+11(x-2)+28]+5

\small =(x-2)^2[x^3(x-2)+3x^2(x-2)+8x(x-2)+21(x-2)+53]+28(x-2)+5.

Thus the integral becomes

\small \int \left \{ x^3+3x^2+8x+21+\frac{53}{x-2}+\frac{28}{(x-2)^2}+\frac{5}{(x-2)^3} \right \} dx

\small =\frac{1}{4}x^4+x^3+4x^2+21x+53ln|x-2|-\frac{28}{(x-2)}-\frac{5}{2(x-2)^2}+C

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