Question

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

Answer 1

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

Answer 2

To integrate the function ∫(x + 7)e^(5x) dx, you can use integration by parts. Integration by parts is a technique based on the product rule for differentiation.

The formula for integration by parts is ∫u dv = uv - ∫v du, where u and dv are chosen parts of the integrand.

In your case, you can choose:

u = (x + 7) => du = dx dv = e^(5x)dx => v = (1/5)e^(5x)

Now, apply the integration by parts formula:

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

Now, you can integrate the remaining term, ∫e^(5x) dx:

∫e^(5x) dx = (1/5)e^(5x) + C

So, the final result is:

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

Now, you can simplify the expression:

(1/5)e^(5x)(x + 7) - (1/25)e^(5x) - C

This is the integral of (x + 7)e^(5x) with the constant of integration (C).