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 , 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 by :
So, the integral becomes:
Step 2: Break the integral into parts
Now we integrate each term separately:
Step 3: Compute each integral
Step 4: Combine the results
The full integral becomes:
Thus, the solution to the integral is:
Would you like more details or have any questions?
Here are 5 related questions to extend your understanding:
- How would the result change if the denominator was instead of ?
- Can you perform the integration of ?
- What happens if you add a constant to the numerator, such as ?
- Can you apply the same method to integrate ?
- How would you integrate 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.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
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
Related Recommendation
Integral of (1 - x + 2x² - x³) / (x(x² + 1)²) using Partial Fraction Decomposition
Integration of Rational Function \( \int \frac{2x}{(x - 1)^2} \, dx \) Using Substitution
Integral of 1 / (2x + 1) - Substitution Method Solution
Integral of (x+4)/(x+1) Using Partial Fraction Decomposition
Integral Transformation for Rational Functions: \(\int \frac{x^2}{x+1} dx\)