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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
Evaluate $ \frac{2x+1}{x^2+x+2}\, dx$
1. $log(2x+1) + c$ where c is an arbitrary constant
2. $log\!\left(\frac{2x+1} {x^2+x+2}\right) + c$ where c is an arbitrary constant
3. $log(x^2+x+2) + c$ where c is an arbitrary constant
4. $log\!\left(\tfrac{1}{2}\right) + c$ where c is an arbitrary constant
Solution
Step 1: Let the denominator be $$f(x) = x^2 + x + 2.$$ Then, $$f'(x) = 2x + 1,$$ which is the numerator.
Step 2: Apply the rule: $$\int \frac{f'(x)}{f(x)} \, dx = \ln|f(x)| + C.$$
Step 3: Therefore, $$\int \frac{2x+1}{x^2+x+2}\, dx = \ln|x^2+x+2| + C.$$
Final Answer: $\log(x^2+x+2) + c$
Correct Option: 3
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