Definite Integration Question 65
Question 65
- The value of $\frac{(5050) \int_{0}^{1}\left(1-x^{50}\right)^{100} d x}{\int_{0}^{1}\left(1-x^{50}\right)^{101} d x}$ is
$(2006,6$ M)
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Solution:
- Let $I_{2}=\int_{0}^{1}\left(1-x^{50}\right)^{101} d x$,
$$ =\left[\left(1-x^{50}\right)^{101} \cdot x\right]{0}^{1}+\int{0}^{1}\left(1-x^{50}\right)^{100} 50 \cdot x^{49} \cdot x d x $$
[using integration by parts]
$$ =0-\int_{0}^{1}(50)(101)\left(1-x^{50}\right)^{100}\left(-x^{50}\right) d x $$
$$ \begin{aligned} = & -(50)(101) \int_{0}^{1}\left(1-x^{50}\right)^{101} d x \ & +(50)(101) \int_{0}^{1}\left(1-x^{50}\right)^{100} d x=5050 I_{2}+5050 I_{1} \end{aligned} $$
$\therefore \quad I_{2}+5050 I_{2}=5050 I_{1}$
$\Rightarrow \quad \frac{(5050) I_{1}}{I_{2}}=5051$
