Integral of cos(ln x) - Calculus Integration by Parts

Published on June 16, 2025

Calculus

This problem involves evaluating the integral of cos(ln x) using integration by parts. The solution involves applying integration by parts twice and solving for the integral.

Question

Answer

The problem asks to evaluate the integral:

We can solve this integral using integration by parts. The formula for integration by parts is

Let

and

.
Then,

.
And

Applying the integration by parts formula:

Now we need to evaluate the integral

. We will use integration by parts again for this integral.
Let

.
Let

and

.
Then,

.
And

Applying the integration by parts formula to

Now substitute this back into the original equation:

Let

Now, we can solve for

Don’t forget the constant of integration,

The final answer is

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