Evaluate the integral.
[tex]\int \frac{x+2}{x^2+5 x-6} d x=[/tex]
Answer (2)
To evaluate the integral ∫ x 2 + 5 x − 6 x + 2 d x , we first factor the denominator and perform partial fraction decomposition. After solving for constants, we rewrite the integral and integrate each term, arriving at the final result: 7 4 ln ∣ x + 6∣ + 7 3 ln ∣ x − 1∣ + C .
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Factor the denominator: x 2 + 5 x − 6 = ( x + 6 ) ( x − 1 ) .
Perform partial fraction decomposition: ( x + 6 ) ( x − 1 ) x + 2 = x + 6 A + x − 1 B , which gives A = 7 4 and B = 7 3 .
Rewrite the integral: ∫ x 2 + 5 x − 6 x + 2 d x = 7 4 ∫ x + 6 1 d x + 7 3 ∫ x − 1 1 d x .
Integrate to get the final answer: 7 4 ln ∣ x + 6∣ + 7 3 ln ∣ x − 1∣ + C .
Explanation
Problem Analysis We are asked to evaluate the integral ∫ x 2 + 5 x − 6 x + 2 d x .
Factor the denominator First, we factor the denominator: x 2 + 5 x − 6 = ( x + 6 ) ( x − 1 ) .
Partial Fraction Decomposition Now, we perform partial fraction decomposition: ( x + 6 ) ( x − 1 ) x + 2 = x + 6 A + x − 1 B Multiplying both sides by ( x + 6 ) ( x − 1 ) , we get: x + 2 = A ( x − 1 ) + B ( x + 6 )
Solve for A and B To solve for A and B , we can use substitution. Let x = 1 :
1 + 2 = A ( 1 − 1 ) + B ( 1 + 6 ) ⇒ 3 = 7 B ⇒ B = 7 3 Let x = − 6 :
− 6 + 2 = A ( − 6 − 1 ) + B ( − 6 + 6 ) ⇒ − 4 = − 7 A ⇒ A = 7 4 So, we have A = 7 4 and B = 7 3 .
Rewrite the integral Now we can rewrite the integral as: ∫ x 2 + 5 x − 6 x + 2 d x = ∫ ( x + 6 7 4 + x − 1 7 3 ) d x = 7 4 ∫ x + 6 1 d x + 7 3 ∫ x − 1 1 d x
Integrate each term Now, we integrate each term: 7 4 ∫ x + 6 1 d x = 7 4 ln ∣ x + 6∣ + C 1 7 3 ∫ x − 1 1 d x = 7 3 ln ∣ x − 1∣ + C 2
Final result Combining the results, we get: ∫ x 2 + 5 x − 6 x + 2 d x = 7 4 ln ∣ x + 6∣ + 7 3 ln ∣ x − 1∣ + C
Examples
Partial fraction decomposition is used in chemical engineering to analyze the composition of mixtures and to design separation processes. For example, when separating a mixture of gases or liquids, engineers use partial fractions to model the concentration of each component as a function of time or distance. This helps in optimizing the design of distillation columns or other separation equipment.
