Evaluate (i)

(ii)

To evaluate the integrals, we’ll tackle them one at a time.

(i) To evaluate , we can use integration by parts. Let:

Now, applying integration by parts:

So, the result is:

(ii) To evaluate the definite integral , we can use substitution. Let:

Now we also need to express in terms of by rearranging:

Now, changing limits according to the substitution:

When , and when , . Plugging everything in:

This simplifies to:

This integral can be solved using the beta function or directly through evaluation techniques. The final answer can be simplified to arrive at a closed form.

Therefore, we find that:

So, the evaluations yield:

(i)

(ii)