Question

Интеграл( 5x -14) /(x^3-x^2-4x+4)

Answer 1

5x-14/x^3-x^2-4x+4
прости что без решение

Answer 2

To find the integral of the given expression, we need to perform partial fraction decomposition and then integrate each term separately. Here's the step-by-step process:

Step 1: Factorize the denominator: The denominator of the expression is a cubic polynomial. We need to factorize it to perform partial fraction decomposition.

x^3 - x^2 - 4x + 4 = (x^2 - 4) * (x - 1) = (x + 2)(x - 2)(x - 1)

Step 2: Perform partial fraction decomposition: We express the given expression as a sum of simpler fractions with undetermined coefficients:

(5x - 14) / (x^3 - x^2 - 4x + 4) = A / (x + 2) + B / (x - 2) + C / (x - 1)

Step 3: Solve for the coefficients A, B, and C: To find the values of A, B, and C, we multiply both sides of the equation by the common denominator (x + 2)(x - 2)(x - 1):

5x - 14 = A(x - 2)(x - 1) + B(x + 2)(x - 1) + C(x + 2)(x - 2)

Now, we can plug in specific values of x to solve for the coefficients.

For x = 2: 5(2) - 14 = A(2 - 2)(2 - 1) + B(2 + 2)(2 - 1) + C(2 + 2)(2 - 2) -4 = 4B B = -1

For x = -2: 5(-2) - 14 = A(-2 - 2)(-2 - 1) + B(-2 + 2)(-2 - 1) + C(-2 + 2)(-2 - 2) -24 = -18A A = 4/3

For x = 1: 5(1) - 14 = A(1 - 2)(1 - 1) + B(1 + 2)(1 - 1) + C(1 + 2)(1 - 2) -9 = -3C C = 3

So the partial fraction decomposition is:

(5x - 14) / (x^3 - x^2 - 4x + 4) = 4/3 / (x + 2) - 1 / (x - 2) + 3 / (x - 1)

Step 4: Integrate each term separately: Now that we have decomposed the integral into simpler fractions, we can integrate each term:

∫(5x - 14) / (x^3 - x^2 - 4x + 4) dx = ∫[4/3 / (x + 2) - 1 / (x - 2) + 3 / (x - 1)] dx

Integrating each term:

∫(4/3) / (x + 2) dx = (4/3) * ln| x + 2 | + C1

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

∫(3) / (x - 1) dx = 3 * ln| x - 1 | + C3

Where C1, C2, and C3 are constants of integration.

Thus, the final integral is:

∫(5x - 14) / (x^3 - x^2 - 4x + 4) dx = (4/3) * ln| x + 2 | - ln| x - 2 | + 3 * ln| x - 1 | + C

where C is the constant of integration.