**Integration of x^(13/2).(1 x^(5/2))^(1/2) with respect to x**

To integrate the given expression, we can use the power rule of integration. The power rule states that the integral of x^n with respect to x is (x^(n+1))/(n+1), where n is any real number except -1.

**Step 1: Simplify the expression**
Let's simplify the expression before integrating it. We have x^(13/2).(1 x^(5/2))^(1/2).

Using the property of exponents, we can rewrite the expression as x^(13/2).(x^(5/2))^1/2.

Simplifying further, we have x^(13/2).x^(5/4).

Using the property of exponents again, we can add the exponents when multiplying, so x^(13/2).x^(5/4) = x^(13/2 + 5/4).

Combining the exponents, we get x^(13/2 + 5/4) = x^(26/4 + 5/4) = x^(31/4).

Therefore, our simplified expression is x^(31/4).

**Step 2: Apply the power rule of integration**
Now that we have simplified the expression, we can integrate it using the power rule.

Using the power rule, the integral of x^(31/4) with respect to x is (x^(31/4 + 1))/(31/4 + 1) = (x^(35/4))/(35/4).

Simplifying further, we have (4/35)x^(35/4).

Hence, the integral of x^(13/2).(1 x^(5/2))^(1/2) with respect to x is (4/35)x^(35/4).

In summary:

∫(x^(13/2).(1 x^(5/2))^(1/2)) dx = (4/35)x^(35/4) + C,

where C is the constant of integration.