Find the general antiderivative

[lovetolearn]

I am not sure if I am doing this correctly or how to really show my work on A.

Find the general antiderivative of the following

b) f(x)=5-4x³+2x^5 / x^6

B) I got
f(x)=(5-4x^3+2x^5)/x^6
f(x)=5x^-6-4x^-3+2x
((5x^-5)/(-5))(-4x^2/-2)+2lnx
= 2ln(x) - x^-5 + 2x^-2
= 2ln(x) - 1/x^5 + 2/x^2
= 2ln(x) + (2x^3 - 1)/x^5

Is this correct?

 

[tkhunny]

There is so much going on in there, I don't know how you decided that you have the expression you intended. Why not one piece at a time?

5-4x³+2x^5 / x^6


5/x^6 ==> ??

-4/x^3 ==> ??

2/x ==> ??

Then reassemble.

By the way, the problem statement asks for the GENERAL antiderivative. This means you must list ALL of them. There are infintely many. What's your plan for this?

 

[soroban]

Hello, lovetolearn!

Find the general antiderivative of the following

. . f(x) .= .(5 - 4x³ + 2x⁵)/x⁶

I got: . f(x) .= .5x^-6 - 4x^-3 + 2/x

Integrate:
. . (5x^-5)/(-5) + (-4x^-2)/(-2) + 2ln(x) + C

. . = . 2ln(x) - x^-5 + 2x^-2 + C

. . = . 2ln(x) - 1/x⁵ + 2/x² + C

. . = . 2ln(x) + (2x³ - 1)/x⁵+ C

Is this correct?

The "plus C" makes it correct . . .