Question

Let $\alpha \in(0,1)$ and $\beta=\log _e(1-\alpha)$ Let $P_n(x)=x+\frac{x^2}{2}+\frac{x^3}{3}+\ldots+\frac{x^n}{n}, x \in(0,1)$ Then the integral $\int\limits_0^\alpha \frac{t^{50}}{1-t} d t$ is equal to

• A. $P_{50}(\alpha)-\beta$
• B. $-\left(\beta+P_{50}(\alpha)\right)$
• C. $\beta+P_{50}(a)$
• D. $\beta-P_{50}(\alpha)$

Answer 1

Answer: (B)

$-\left(\beta+P_{50}(\alpha)\right)$

Answer 2

The correct answer is (B) : $-\left(\beta+P_{50}(\alpha)\right)$
0∫α​1−tt50−1+1​=−0∫α​(1+t+…..+t49)+0∫α​1−t1​dt
=−(50α50​+49α49​+…..+1α1​)+(−1ln(1−f)​)0α​
=−P50​(α)−ln(1−α)
=−P50​(α)−β