Question

Find the indefinite integral of xe⁵x dx. (Note: Solve by the simplest method–not all require integration by parts. Use c for the constant of integration.)

Answer 1

To find the indefinite integral of xe⁵x dx, we can use integration by parts. Let u = x and dv = e⁵x dx. Then, du = dx and v = ∫e⁵x dx. Using the formula for integration by parts, ∫u dv = uv - ∫v du, we can now calculate the integral.

Step-by-step explanation:

To find the indefinite integral of xe⁵x dx, we can use integration by parts. Let u = x and dv = e⁵x dx. Then, du = dx and v = ∫e⁵x dx. Using the formula for integration by parts, ∫u dv = uv - ∫v du, we can now calculate the integral.

1. Let u = x and dv = e⁵x dx.
2. Calculate du = dx and v = ∫e⁵x dx.
3. Using the formula ∫u dv = uv - ∫v du, we get ∫xe⁵x dx = uv - ∫v du.
4. Substituting the values, we have ∫xe⁵x dx = x(∫e⁵x dx) - ∫(∫e⁵x dx) dx.
5. Simplifying further, ∫xe⁵x dx = x(∫e⁵x dx) - ∫e⁵x dx + C.
6. Thus, the indefinite integral of xe⁵x dx is x(∫e⁵x dx) - ∫e⁵x dx + C.