1.3. $\int_{0}^{\pi / 2}\left(\int_{0}^{\sin y}(x \cos y+1) d x\right) d y$

Step 1 ：First, we need to solve the inner integral. The integral of \(x\cos(y)\) with respect to x is \(0.5x^2\cos(y)\), and the integral of 1 with respect to x is x. We evaluate these at the limits of the inner integral, which are 0 and \(\sin(y)\).

Step 2 ：Then, we solve the outer integral. The integral of \(0.5\sin^2(y)\cos(y)\) with respect to y is a bit complicated, but can be solved using trigonometric identities. The integral of \(\sin(y)\) with respect to y is \(-\cos(y)\). We evaluate these at the limits of the outer integral, which are 0 and \(\pi/2\).

Step 3 ：Finally, we subtract the lower limit from the upper limit to get the final answer. The final answer is \(\boxed{\frac{7}{6}}\).