Question

∫ X^3/(X^2+4)^2dx
∫ X^3/(X^2+4)^2dx @Mathematics

Answer 1

\[\int \frac{x^3}{(x^2+4)^2}dx\]
yes?

Answer 2

\[\frac{x^3}{(x^2+4)^2}=\frac{Ax+B}{x^2+4}+\frac{Cx+D}{(x^2+4)^2}=\frac{(Ax+B)(x^2+4)+(Cx+D)}{(x^2+4)^2}\]

Answer 3

=>
\[x^3=Ax^3+4Ax+Bx^2+4B+Cx+D\]

Answer 4

\[x^3=Ax^3+Bx^2+x(4A+C)+(4B+D)\]

Answer 5

\[=>A=1, B=0, 4A+C=0, 4B+D=0\]

Answer 6

=>C=-4, D=0

Answer 7

\[\int\limits_{}^{}(\frac{1x}{x^2+4}+\frac{-4x}{(x^2+4)^2}) dx\]

Answer 8

this should be easy now for you
just use a substitution like u=x^2+4 =>du=2x dx

Answer 9

what happens if you try
\[u=x^2+4\]
\[du=2xdx\]
\[x^2=u-4\] ? this is an honest question because i am not sure

Answer 10

yea i got it from here thank you

Answer 11

dont think substitution would work cuz the top is X^3

Answer 12

maybe not but let me try anyway. i know we have
\[x^3dx\] but if i use
\[\frac{1}{2}du=xdx\] all i need in the top is
\[x^2=u-4\]

Answer 13

are you asking is there a way to avoid writing it by partial fractions?

Answer 14

yes i am just wondering (not being a critic)

Answer 15

you could use a trig substitution

Answer 16

what about
\[\frac{1}{2}\int\frac{u-4}{u^2}du\] do we get the same answer?

Answer 17

yep that looks like it works too gj

Answer 18

\[\frac{1}{2}[\int\frac{1}{u}du-\int\frac{4}{u^2}du]\]
\[\frac{1}{2}\ln(u)+\frac{2}{u}\]

Answer 19

\[\frac{1}{2}\ln(x^2+4)+\frac{2}{x^2+4}\] is that the same?

Answer 20

i bet so
i see nothing wrong with your way

Answer 21

+C

Answer 22

not they are certainly the same

Answer 23

now*

Answer 24

yay!

Answer 25

now if i don't get some actual work done i am going to be in trouble

Answer 26

yep i need to read a paper

Answer 27

\[\int \frac{x^3}{(x^2+4)^2}dx\]
\[\int x^2\frac{x}{(x^2+4)^2}dx\]
\[\frac{1}{2}\int x^2\frac{2x}{(x^2+4)^2}dx\]
\[\frac{1}{2}(-\frac{x^2}{x^2+4}-\int \frac{2x}{x^2+4}dx)\]
\[-\frac{x^2}{2(x^2+4)}-\frac{1}{2}\int \frac{2x}{x^2+4}dx\]
\[-\frac{x^2}{2(x^2+4)}-\frac{1}{2}ln|x^2+4|+C\]
whered i mess up ?

Answer 28

i dropped a negative ... i see that

Answer 29

more than one way to skin a cat