jee-main 2023 Q75

jee-main · India · session1_31jan_shift1 Integration by Substitution Substitution to Prove an Integral Identity or Equality

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 $\int_0^{\alpha} \frac{t^{50}}{1-t}\,dt$ is equal to
(1) $\beta - P_{50}(\alpha)$
(2) $-\beta + P_{50}(\alpha)$
(3) $P_{50}(\alpha) - \beta$
(4) $\beta + P_{50}(\alpha)$

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 $\int_0^{\alpha} \frac{t^{50}}{1-t}\,dt$ is equal to\\
(1) $\beta - P_{50}(\alpha)$\\
(2) $-\beta + P_{50}(\alpha)$\\
(3) $P_{50}(\alpha) - \beta$\\
(4) $\beta + P_{50}(\alpha)$

Paper Questions

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