The integral in the image is:

$\int_0^{1+i} z \, dz \text{ along the line } y = x.$

Step 1: Parametrize the path

Since the line is $y = x$, the path from $z = 0$ to $z = 1 + i$ can be parametrized as: $z(t) = t + it, \quad \text{where} \quad t \in [0, 1].$

So, the variable $z$ changes linearly along this line.

Step 2: Compute $dz$

Differentiating the parametrization with respect to $t$: $dz = (1 + i) \, dt.$

Step 3: Substitute into the integral

Now substitute $z(t) = t(1 + i)$ and $dz = (1 + i) \, dt$ into the integral: $\int_0^{1+i} z \, dz = \int_0^1 t(1+i)(1+i) \, dt.$

Step 4: Simplify the integrand

We first calculate $(1+i)(1+i)$: $(1+i)(1+i) = 1 + 2i - 1 = 2i.$

Thus, the integrand becomes: $\int_0^1 t \cdot 2i \, dt = 2i \int_0^1 t \, dt.$

Step 5: Solve the integral

The integral of $t$ is straightforward: $\int_0^1 t \, dt = \left[ \frac{t^2}{2} \right]_0^1 = \frac{1^2}{2} = \frac{1}{2}.$

So, the integral becomes: $2i \cdot \frac{1}{2} = i.$

Final Answer:

The value of the integral is $\boxed{i}$.

Would you like further details on any step or have additional questions?

Here are some related questions:

1. What is the geometric interpretation of complex line integrals?
2. How do you parametrize different paths in the complex plane?
3. How would the integral change if the path was along a curve instead of a line?
4. What are the general steps for computing contour integrals in complex analysis?
5. How does the result change if we modify the integral limits?

Tip: In complex integrals, always check if the path is straight or curved to correctly parametrize the variable.