Integral de una división de polinomios (paso a paso)

Summary

In this tutorial video, the process of calculating the integral of a rational function is explained step-by-step. The video demonstrates how to divide polynomials, using the division of `x^2 + 8x - 7` by `x - 3`, resulting in a simplified expression of `x + 11 + 26/(x - 3)`. The integral is then broken into three simpler parts: ∫x dx, ∫11 dx, and ∫(26/(x - 3)) dx, which are solved to give the final result: (x^2)/2 + 11x + 26ln|x - 3| + C. The video encourages viewers to like, subscribe, and engage with questions or comments.

Takeaways

• 😀 The video teaches how to calculate the integral of a polynomial expression divided by another polynomial, specifically 'x^3 + 8x - 7' divided by 'x - 3'.
• 😀 The process starts with polynomial division since the degree of the numerator is higher than that of the denominator.
• 😀 The degree of a polynomial is determined by the highest exponent of the variable, and in this case, the numerator's degree is 3, while the denominator's degree is 1.
• 😀 To perform the division, the first term of the numerator is divided by the first term of the denominator, resulting in the term 'x'.
• 😀 After dividing, the result is multiplied by the denominator, and the sign is changed to subtract it from the original expression.
• 😀 The process is repeated by dividing the next term of the remainder by the denominator, continuing the division until no further terms can be divided.
• 😀 The division results in 'x + 11', with a remainder of '26/(x - 3)', which must be included in the final result of the division.
• 😀 The integral is simplified into three parts: the integral of 'x', the integral of '11', and the integral of '26/(x - 3)'.
• 😀 Constants multiplying the integrals (like 11 and 26) are factored out to simplify the calculations.
• 😀 The final integrals are solved using standard integration rules: the integral of 'x' results in 'x^2/2', the integral of '11' gives '11x', and the integral of '26/(x - 3)' gives '26 * ln|x - 3|'.
• 😀 The constant of integration is added at the end, and the result is finalized as 'x^2/2 + 11x + 26 * ln|x - 3| + C'.

Q & A

What is the integral that needs to be calculated in the video?

The integral that needs to be calculated is: ∫ (x^3 + 8x - 7) / (x - 3) dx.

Why is polynomial division necessary in this problem?

Polynomial division is necessary because the degree of the numerator (x^3 + 8x - 7) is higher than that of the denominator (x - 3), requiring division to simplify the expression.

How do you determine the degree of a polynomial?

The degree of a polynomial is determined by the highest exponent of the variable in the polynomial. For example, in the polynomial x^3 + 8x - 7, the degree is 3 because the highest exponent is 3.

What is the result of the polynomial division in this example?

The result of the polynomial division is: x^2 + 11 + 26 / (x - 3).

What is the next step after performing polynomial division?

After performing polynomial division, the integral is rewritten as the sum of simpler integrals: ∫ (x^2 + 11 + 26 / (x - 3)) dx.

How do you separate the integrals after rewriting the expression?

The expression is separated into three simpler integrals: ∫ x^2 dx, ∫ 11 dx, and ∫ 26 / (x - 3) dx.

What is the result of the integral ∫ x^2 dx?

The result of the integral ∫ x^2 dx is (x^3 / 3).

What happens when you integrate a constant like 11 with respect to x?

When you integrate a constant like 11 with respect to x, the result is simply the constant multiplied by x. In this case, ∫ 11 dx = 11x.

How do you integrate the expression ∫ 26 / (x - 3) dx?

The integral ∫ 26 / (x - 3) dx is a standard logarithmic integral, and the result is 26 ln |x - 3|.

What is the final result of the integral calculation?

The final result of the integral calculation is: (x^3 / 3) + 11x + 26 ln |x - 3| + C, where C is the constant of integration.