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JEE Advanced 2006
Exam
JEE Advanced
Year
2006
Subject
Mathematics
Chapter
Definite Integration

JEE Advanced 2006 · Mathematics · Definite Integration

The value of (5050) ∫_0^1(1-x^ 50 )^ 100 d x ∫_0^1(1-x^ 50 )^ 101 d...

The value of (5050)∫01(1−x50)100dx∫01(1−x50)101dx\frac{(5050) \int_0^1\left(1-x^{50}\right)^{100} d x}{\int_0^1\left(1-x^{50}\right)^{101} d x}

Correct answer

0

Step-by-step solution

Let I2=∫01(1−x50)101dxI_2=\int_0^1\left(1-x^{50}\right)^{101} d x, using integration by parts =[(1−x)50)101\cdotx]01+∫01(1−x50)10050\cdotx49\cdotxdx=0−∫01(50)(101)(1−x50)100(−x)50dx=−(50)(101)∫01(1−x50)101dx+(50)(101)∫01(1−x50)100dx=5050I2+5050I1\begin{aligned}& \left.=\left[(1-x)^{50}\right)^{101} \cdotx\right]_0^1+\int_0^1\left(1-x^{50}\right)^{100} 50 \cdotx^{49} \cdotx d x \\ & =0-\int_0^1(50)(101)\left(1-x^{50}\right)^{100}(-x)^{50} d x \\ & =-(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}∴I2+5050I2=5050I1∴(5050)I1I2=5051\begin{aligned} \therefore& I_2+5050 I_2 & =5050 I_1 \\ \therefore & \frac{(5050) I_1}{I_2}& =5051 \end{aligned}
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