Integrate: ∫ e^x sin x d x.

To solve the integral ∫ e^x sin x dx, we can use the method of integration by parts or the technique of integration using complex numbers. Here, we will use integration by parts.

Step-by-Step Solution

First, we will apply integration by parts twice. We start by letting:

• u = sin x and dv = e^x dx

Then, we differentiate and integrate:

• du = cos x dx
• v = e^x

Now, we apply the integration by parts formula:

∫ u dv = uv - ∫ v du

Substituting our values, we get:

∫ e^x sin x dx = e^x sin x - ∫ e^x cos x dx

Second Integration by Parts

Next, we need to solve the integral ∫ e^x cos x dx. We apply integration by parts again:

• u = cos x and dv = e^x dx

Then, we differentiate and integrate:

• du = -sin x dx
• v = e^x

Applying the formula again:

∫ e^x cos x dx = e^x cos x + ∫ e^x sin x dx

Combining Results

Now, we substitute this back into our earlier equation:

∫ e^x sin x dx = e^x sin x - (e^x cos x + ∫ e^x sin x dx)

Rearranging gives:

∫ e^x sin x dx + ∫ e^x sin x dx = e^x sin x - e^x cos x

This simplifies to:

2∫ e^x sin x dx = e^x (sin x - cos x)

Final Result

Dividing both sides by 2, we find:

∫ e^x sin x dx = 1/2 e^x (sin x - cos x) + C

where C is the constant of integration. This is the final answer for the integral.