integrate (x + 7) * e ^ (5x) dx

To integrate the function ∫(x + 7)e^(5x) dx, we can use integration by parts. The formula for integration by parts is:

∫u * dv = uv - ∫v * du

Let's assign u = (x + 7) and dv = e^(5x) dx. Then, we can differentiate u to find du and integrate dv to find v.

Differentiating u:

du = dx

Integrating dv:

∫e^(5x) dx = (1/5)e^(5x)

Now we can apply the integration by parts formula:

∫(x + 7)e^(5x) dx = u * v - ∫v * du = (x + 7) * (1/5)e^(5x) - ∫(1/5)e^(5x) dx

Simplifying the expression, we have:

∫(x + 7)e^(5x) dx = (x + 7) * (1/5)e^(5x) - (1/5)∫e^(5x) dx

Integrating the remaining term:

∫e^(5x) dx = (1/5)e^(5x)

Substituting this back into the equation:

∫(x + 7)e^(5x) dx = (x + 7) * (1/5)e^(5x) - (1/5)(1/5)e^(5x) + C

where C is the constant of integration.

Therefore, the integral of (x + 7)e^(5x) dx is:

∫(x + 7)e^(5x) dx = (x + 7) * (1/5)e^(5x) - (1/25)e^(5x) + C