Indefinite Integration 1 Question 65
66. 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]
$$ =0-\int _0^{1}(50)(101)\left(1-x^{50}\right)^{100}\left(-x^{50}\right) d x $$
$$ \begin{aligned} = & -(50)(101) \int _0^{1}\left(1-x^{50}\right)^{101} d x \\ & +(50)(101) \int _0^{1}\left(1-x^{50}\right)^{100} d x=5050 I _2+5050 I _1 \end{aligned} $$
$\therefore \quad I _2+5050 I _2=5050 I _1$
$\Rightarrow \quad \frac{(5050) I _1}{I _2}=5051$
