Question

Интеграл 3x^2dx/(2-x^3)^4 Посчитать Методом замены переменной

Answer 1

Решение в приложении.

Answer 2

Calculation of the Integral

To calculate the integral ∫(3x^2dx)/(2-x^3)^4 using the method of variable substitution, we can follow these steps:

1. Let's start by making a substitution: u = 2 - x^3. This will help simplify the integral.

2. Next, we need to find the derivative of u with respect to x. Taking the derivative of both sides of the equation u = 2 - x^3, we get du/dx = -3x^2.

3. Now, we can express dx in terms of du using the derivative we just found: dx = du/(-3x^2).

4. Substituting the values of u and dx in the integral, we have:

∫(3x^2dx)/(2-x^3)^4 = ∫(3x^2)(du/(-3x^2))/(2-x^3)^4 Simplifying, we get: ∫(-du)/(2-x^3)^4

5. The integral now becomes: ∫(-du)/(2-u)^4.

6. To solve this integral, we can use the power rule for integration. The integral of (-du)/(2-u)^4 is equal to (1/3)(2-u)^-3 + C, where C is the constant of integration.

7. Finally, substituting back u = 2 - x^3, we get the final result:

∫(3x^2dx)/(2-x^3)^4 = (1/3)(2 - (2 - x^3))^-3 + C Simplifying further, we have: (1/3)(x^3 - 2)^-3 + C.

Therefore, the integral of (3x^2dx)/(2-x^3)^4, using the method of variable substitution, is (1/3)(x^3 - 2)^-3 + C.

Please note that this is a general solution, and if you have specific limits of integration, you can substitute those values to find the definite integral.