Integrating tan x ln x cos x: A Guide to Solving Indefinite Integrals

[James889]

Hi,

Looking to integrate the indefinite integral:

$$\int tan~x\cdot ln~x\cdot cos~x$$

Since tan x = sin x/ cos x, this integral be written as $$\int sin~x\cdot ln~x$$

In that case i thought the answer was cot x. But that is wrong.

Do you need to use integration by parts on this one?

 

[3029298]

I just tried to evaluate this integral in Mathematica, but this does not return an analytical solution... so I doubt there is one. But maybe someone else has an idea?

 

[Altabeh]

Hi,

Looking to integrate the indefinite integral:

$$\int tan~x\cdot ln~x\cdot cos~x$$

Since tan x = sin x/ cos x, this integral be written as $$\int sin~x\cdotln~x$$

In that case i thought the answer was cot x. But that is wrong.

Do you need to use integration by parts on this one?

The integral sounds so unco and tricky at the first glance. But later you'll find out that it is equivalent to $$\int sin~x\cdot ln~x\ dx$$ after cancelling the cos(x) with that of tangant function. So letting f'(x) = sin(x) and g(x) = ln(x) and using integration by parts gives us $$-\cos \left( x \right) \ln \left( x \right) +{\it Ci} \left( x\right)$$. Note that here we assume that x is greater than zero. otherwise the above antiderivative will get some complex term. At x=0, it is not defined.

Here Ci(x) is some kind of special funcion called Cosine Integral which is exactly the second part in the integration you are supposed to do.

AB