Question

Вычислить интеграл xe^x dx методом интегрирования по частям

Answer 1

$\int xe^x \ dx = \int x \ de^x = xe^x - \int e^x \ dx = xe^x - e^x + C$

Можно ответ записать так:

$e^x(x - 1) + C$

Answer 2

Calculating the Integral of xe^x dx using Integration by Parts

To calculate the integral ∫xe^x dx using the method of integration by parts, we can follow these steps:

Step 1: Identify the functions u(x) and v'(x) in the integral. - Let u(x) = x - Let v'(x) = e^x

Step 2: Calculate the derivatives of u(x) and the antiderivative of v'(x). - Differentiating u(x), we get du = dx - Integrating v'(x), we get v(x) = e^x

Step 3: Apply the integration by parts formula. The integration by parts formula is given by: ∫u(x)v'(x) dx = u(x)v(x) - ∫v(x)u'(x) dx

Using this formula, we can calculate the integral of xe^x dx.

∫xe^x dx = u(x)v(x) - ∫v(x)u'(x) dx = x * e^x - ∫e^x dx

Step 4: Evaluate the remaining integral. The integral of e^x dx is simply e^x.

∫xe^x dx = x * e^x - e^x + C

Therefore, the integral of xe^x dx is x * e^x - e^x + C, where C is the constant of integration.

Please let me know if you need any further clarification or assistance!