integrate: ((x^4-2x)/(x+4))?
i think this can be done with synthetic division.. but not sure. please help
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Yes it can… I will try to write it out but it might get messy…
““`_____x^3-4x²+16x-66
x-4)x^4+0x^3+0x-2x
“`- x^4+4x______
““““`_-4x^3+0x²
“““““`-4x^3-16x²______
“““““““““_16x²-2x
““““““““““16x²+64x_____
““““““““““““`_-66x+0
“““““““““““““`-66x-264
So…(x^4-2x)/(x+4) = x^3-4x²+16x-66 + 264/x+4
∫(x^3-4x²+16x-66)dx + ∫(264/x+4)dx –> u=x+4 du=dx
x^4/4-4x^3/3+8x²-66x + ∫(264/u)du
x^4/4-4x^3/3+8x²-66x + 264∫(1/u)du
x^4/4-4x^3/3+8x²-66x + 264ln|x+4| +C
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put y = x+4, then, x= y-4 and dx=dy
Now ur problem becomes
{(y-4)^4 -2(y-4)}dy/(y)
expand the series, divide each term by y, and integrate separate term…
After you get the answer in terms of y, re-substitute y= x+4 to get the final answer
Hope that helps
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