Integration Of Rational Functions

Concept: Use trigonometric substitution: This simplifies expressions of the form .
Step 1: Substitute Let: Then,
Step 2: Transform the integral
Step 3: Use identity
Step 4: Simplify Using:

Step 5: Back-substitute Since: So, Substitute back: Final Result: Explanation: Trigonometric substitution converts the radical into a simple trigonometric integral, which is then transformed back into algebraic form.

Concept: When the numerator resembles the derivative of the denominator, use substitution: If not exact, split the numerator appropriately.
Step 1: Let Differentiate denominator: We express numerator in terms of :
Step 2: Split the integral
Step 3: First integral (log form)
Step 4: Second integral (complete square) So, Using:

Step 5: Multiply constant Final Answer: Explanation: The numerator was decomposed into a derivative form plus a constant. One part gave a logarithm, while the remaining quadratic produced an inverse tangent after completing the square.