JEE Mains · Maths · STD 12 - 7.1 indefinite integral
\(\int \frac{\left(x^{2}+1\right) e^{x}}{(x+1)^{2}} d x=f(x) e^{x}+C\), Where \(C\) is a constant, then \(\frac{d^{3} f}{d x^{3}}\) at \(x =1\) is equal to
- A \(\frac{3}{4}\)
- B \(-\frac{3}{4}\)
- C \(-\frac{3}{2}\)
- D \(\frac{3}{2}\)
Answer & Solution
Correct Answer
(A) \(\frac{3}{4}\)
Step-by-step Solution
Detailed explanation
\(\int\left(\frac{x^{2}+1}{(x+1)^{2}}\right) e^{x} \cdot d x\) \(=\int\left(\frac{x^{2}-1+2}{(x+1)^{2}}\right) e^{x} d x\) \(=\int\left(\frac{x-1}{x+1}+\frac{2}{(x+1)^{2}}\right) e^{x} d x\) \(=\int\left(f(x)+f^{\prime}(x)\right) e^{x} d x\) \(=f(x) e^{x}+c\) Where…
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