Math Problem Statement

Use the formula below or any other method for the integration of rational functions in Q1. ∫(ax+b)/(cx+d) dx=a/cx + (bc-ad)/c^2 ln│cx + d│+ c Evaluate: ∫x^2/(x+1) dx

Solution

To evaluate the integral x2x+1dx\int \frac{x^2}{x+1} \, dx, we will first simplify the expression using polynomial division, since the degree of the numerator is higher than the degree of the denominator.

Step 1: Polynomial Division

We divide x2x^2 by x+1x+1:

x2x+1=x1+1x+1\frac{x^2}{x+1} = x - 1 + \frac{1}{x+1}

So, the integral becomes: x2x+1dx=(x1+1x+1)dx\int \frac{x^2}{x+1} \, dx = \int \left( x - 1 + \frac{1}{x+1} \right) dx

Step 2: Break the integral into parts

Now we integrate each term separately: (x1+1x+1)dx=xdx1dx+1x+1dx\int \left( x - 1 + \frac{1}{x+1} \right) dx = \int x \, dx - \int 1 \, dx + \int \frac{1}{x+1} \, dx

Step 3: Compute each integral

  • xdx=x22\int x \, dx = \frac{x^2}{2}
  • 1dx=x\int 1 \, dx = x
  • 1x+1dx=lnx+1\int \frac{1}{x+1} \, dx = \ln |x+1|

Step 4: Combine the results

The full integral becomes: x22x+lnx+1+C\frac{x^2}{2} - x + \ln |x+1| + C

Thus, the solution to the integral is: x2x+1dx=x22x+lnx+1+C\int \frac{x^2}{x+1} \, dx = \frac{x^2}{2} - x + \ln |x+1| + C


Would you like more details or have any questions?

Here are 5 related questions to extend your understanding:

  1. How would the result change if the denominator was x+2x+2 instead of x+1x+1?
  2. Can you perform the integration of x3x+1dx\int \frac{x^3}{x+1} \, dx?
  3. What happens if you add a constant kk to the numerator, such as x2+kx^2 + k?
  4. Can you apply the same method to integrate x2+2x+1x+1dx\frac{x^2 + 2x + 1}{x+1} \, dx?
  5. How would you integrate xx+1\frac{x}{x+1} directly without polynomial division?

Tip: Always check the degree of the numerator and denominator when integrating rational functions. If the degree of the numerator is larger, polynomial division can simplify the integral.

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Math Problem Analysis

Mathematical Concepts

Integration
Rational Functions
Polynomial Division

Formulas

∫(ax+b)/(cx+d) dx = a/cx + (bc-ad)/c^2 ln|cx+d| + C
∫x^2/(x+1) dx = ∫(x - 1 + 1/(x+1)) dx

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Undergraduate Calculus or Advanced High School