Question

∫x^3/(x^2+2x-1 )dx ............

Answer 1

Ответ:

Чтобы интегрировать это выражение, мы можем использовать u-замещение. Пусть u = x^2 + 2x - 1. Тогда du/dx = 2x + 2, или (1/2)du/dx = x + 1.

Answer 2

To evaluate the integral ∫(x^3)/(x^2+2x-1) dx, we can use partial fraction decomposition and then integrate each term separately. Here's how we can proceed:

1. Factorize the denominator (x^2 + 2x - 1) into two binomial factors. To do this, we need to find two numbers that multiply to give -1 and add up to 2. The numbers are 1 and -1.

   x^2 + 2x - 1 = (x + 1)(x - 1)

2. Now, we can express the fraction as the sum of two partial fractions:

   (x^3)/(x^2 + 2x - 1) = A/(x + 1) + B/(x - 1)

   To find the values of A and B, we need to determine the numerators of the partial fractions. Multiply the entire equation by the denominator (x^2 + 2x - 1):

   x^3 = A(x - 1) + B(x + 1)

3. Expand and equate coefficients of like terms:

   x^3 = (A + B)x + (-A + B)

   Equating the coefficients of x^3: 1 = A + B

   Equating the coefficients of x: 0 = -A + B

   Solving these equations, we find A = -1/2 and B = 3/2.

4. Now we can rewrite the integral as:

   ∫(x^3)/(x^2 + 2x - 1) dx = ∫(-1/2)/(x + 1) dx + ∫(3/2)/(x - 1) dx

5. Integrating each term separately:

   ∫(-1/2)/(x + 1) dx = -1/2 ln|x + 1| + C1

   ∫(3/2)/(x - 1) dx = 3/2 ln|x - 1| + C2

   Here, C1 and C2 are constants of integration.

6. Combining the results:

   ∫(x^3)/(x^2 + 2x - 1) dx = -1/2 ln|x + 1| + 3/2 ln|x - 1| + C

   where C is the constant of integration.

Therefore, the integral of (x^3)/(x^2 + 2x - 1) dx is -1/2 ln|x + 1| + 3/2 ln|x - 1| + C, where C is the constant of integration.