Question

Been trying to solve this tricky integral for a while, any help would be appreciated:
Integrate (cosx+xsinx)/x(x+cosx)

Answer 1

I tried using wolfram alpha, it does give a final answer for integral, but does not provide the steps. I tried doing a few substitutions but it doesn't seem to be working out.

Answer 2

ok,whats the derivative of (x+cos x) ?

Answer 3

@battlefire44

Answer 4

Sorry back

Answer 5

Deriv of x + cosx is 1 -sinx, are you suggesting we can do something with this?

Answer 6

yup
u know this:
\[\int\limits_{}^{}f \prime (x)/f(x)=\ln|f(x)|+c \]

Answer 7

Hmm, can you give a quick example, not sure if i seen that generical formula before, so if we have f(x) = x² then 2x/x² = 2/x and hence integral is 2lnx + c ?

Answer 8

yup.

Answer 9

Does that formula have a specefic name, i could look up a proof for it later, but ill take it as a given for now.

Answer 10

u can find that in any complete formula list for integration
going ahead with question,
i will write numerator as:
(x+cos x)-x+x sin x
because i have (x+cos x) in denominator
ok??

Answer 11

Yep

Answer 12

that will be
(x+cos x)-x(1-sin x)
ok?
now separate the numerator and tell me what u get.

Answer 13

yep

Answer 14

Appologize for leaving, accidently closed the site, but we will get 1/x + (1-sinx)/(1+cosx)

Answer 15

\[\int\limits x^{2} dx = 2\ln x + c ???\]

Answer 16

and then we can integrate seperately, and arrive at answer, i got it

Answer 17

thats correct,tell me final answer.

Answer 18

lnx + ln(1+cosx) + c

Answer 19

ln |x|-ln|x+cos x|+c

Answer 20

srry i meant, lnx + ln(x+cosx) + c

Answer 21

minus.

Answer 22

yep, thanks soo much, are you a math student?

Answer 23

oops, forgot the minus, but i get it

Answer 24

welcome :) i am an enginneering student.

Answer 25

ohhh, great, thanks soo much :)