How to integrate x^3*e^x?

To integrate x^3*e^x, use integration by parts.

Integration by parts is a technique used to integrate the product of two functions. It involves choosing one function to differentiate and the other to integrate. The formula for integration by parts is:

∫u dv = uv - ∫v du

where u and v are functions of x, and du/dx and dv/dx are their respective derivatives.

To integrate x^3*e^x, let u = x^3 and dv/dx = e^x. Then, du/dx = 3x^2 and v = e^x. Substituting these values into the formula, we get:

∫x^3*e^x dx = x^3*e^x - ∫e^x * 3x^2 dx

The integral on the right-hand side can be evaluated using integration by parts again. Let u = 3x^2 and dv/dx = e^x. Then, du/dx = 6x and v = e^x. Substituting these values into the formula, we get:

∫x^3*e^x dx = x^3*e^x - e^x * 3x^2 + ∫e^x * 6x dx

The integral on the right-hand side can be evaluated using integration by parts one more time. Let u = 6x and dv/dx = e^x. Then, du/dx = 6 and v = e^x. Substituting these values into the formula, we get:

∫x^3*e^x dx = x^3*e^x - e^x * 3x^2 + e^x * 6x - ∫e^x * 6 dx

The final integral can be evaluated as ∫e^x * 6 dx = 6e^x + C, where C is the constant of integration. Therefore, the final answer is:

∫x^3*e^x dx = x^3*e^x - e^x * 3x^2 + e^x * 6x - 6e^x + C