SATHEE: Indefinite Integration 1 Question 65

Indefinite Integration 1 Question 65

66. The value of $\frac{(5050) \int_{0}^{1} {(1 - x^{50})}^{100} d x}{\int_{0}^{1} {(1 - x^{50})}^{101} d x}$ is

$(2006, 6$ M)

Show Answer

Solution:

Let $I_{2} = \int_{0}^{1} {(1 - x^{50})}^{101} d x$ ,

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

[using integration by parts]

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

$\begin{aligned} = & - (50) (101) \int_{0}^{1} {(1 - x^{50})}^{101} d x \\ + (50) (101) \int_{0}^{1} {(1 - x^{50})}^{100} d x = 5050 I_{2} + 5050 I_{1} \end{aligned}$

$∴ I_{2} + 5050 I_{2} = 5050 I_{1}$

$\Rightarrow \frac{(5050) I_{1}}{I_{2}} = 5051$

Indefinite Integration 1 Question 65

66. The value of $\frac{(5050) \int_{0}^{1} {(1 - x^{50})}^{100} d x}{\int_{0}^{1} {(1 - x^{50})}^{101} d x}$ is

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Indefinite Integration 1 Question 65

66. The value of (5050)∫01(1−x50)100dx∫01(1−x50)101dx is

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

66. The value of $\frac{(5050) \int_{0}^{1} {(1 - x^{50})}^{100} d x}{\int_{0}^{1} {(1 - x^{50})}^{101} d x}$ is