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JEE Advanced · 2006 · 09 Apr 2006 · Definite Integrals

The value of

5050 \frac{\int _{0}^{1} ( 1 - x ^{50} ) ^{100} d x}{\int _{0}^{1} ( 1 - x ^{50} ) ^{101} d x}

is

Answer: 5051

Solution

Let

I_{n} = \int_{0}^{1} (1 - x^{50})^{n} d x

.

Using integration by parts on

I_{101}

:

I_{101} = \int_{0}^{1} (1 - x^{50}) (1 - x^{50})^{100} d x = I_{100} - \int_{0}^{1} x^{50} (1 - x^{50})^{100} d x

For the second integral, substitute

u = 1 - x^{50}

,

d u = - 50 x^{49} d x

:

\int_{0}^{1} x \cdot x^{49} (1 - x^{50})^{100} d x = [- \frac{x ( 1 - x ^{50} ) ^{101}}{5050}]_{0}^{1} + \int_{0}^{1} \frac{( 1 - x ^{50} ) ^{101}}{5050} d x

= 0 + \frac{I _{101}}{5050}

Therefore:

I_{101} = I_{100} - \frac{I _{101}}{5050}

I_{101} (1 + \frac{1}{5050}) = I_{100}

\frac{5051}{5050} I_{101} = I_{100}

5050 \frac{I _{100}}{I _{101}} = 5051

The answer is 5051.

JEE Advanced · 2006 · 09 Apr 2006 · Numerical answer

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