# An indefinite integral

1. Jan 7, 2010

### 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?

Last edited: Jan 7, 2010
2. Jan 7, 2010

### 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?

3. Jan 7, 2010

### Altabeh

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.

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