SATHEE: Indefinite Integration 1 Question 90

Indefinite Integration 1 Question 90

91. The value of $\int_{0}^{1} 4 x^{3} \frac{d^{2}}{d x^{2}} {(1 - x^{2})}^{5} d x$ is

(2014 Adv.)

Show Answer

Answer:

Correct Answer: 91. (2)

Solution:

PLAN Integration by parts

$\int f (x) g (x) d x = f (x) \int g (x) d x - \int \frac{d}{d x} [f (x)] \int g (x) d x d x$

Given, $I = \int_{0}^{1} 4 x^{3} \frac{d^{2}}{d x^{2}} {(1 - x^{2})}^{5} d x$

$= 4 x^{3} \frac{d}{d x} {(1 - x^{2})}^{5}_{0}^{1} - \int_{0}^{1} 12 x^{2} \frac{d}{d x} {(1 - x^{2})}^{5} d x$

$\begin{aligned} = 4 x^{3} \times 5 {(1 - x^{2})}^{4} (- 2 x) \\ - 12 {[x^{2} {(1 - x^{2})}^{5}]}_{0}^{1} - \int_{0}^{1} 2 x {(1 - x^{2})}^{5} d x \\ = 0 - 0 - 12 (0 - 0) + 12 \int_{0}^{1} 2 x {(1 - x^{2})}^{5} d x \\ = 12 \times - \frac{{(1 - x^{2})}^{6}}{6}_{0}^{1} = 120 + \frac{1}{6} = 2 \end{aligned}$

Play Store

App Store

Ask SATHEE

Welcome to SATHEE !
Select from 'Menu' to explore our services, or ask SATHEE to get started. Let's embark on this journey of growth together! 🌐📚🚀🎓

I'm relatively new and can sometimes make mistakes.
If you notice any error, such as an incorrect solution, please use the thumbs down icon to aid my learning.
To begin your journey now, click on "I understand".

Indefinite Integration 1 Question 90

91. The value of ∫014x3d2dx2(1−x2)5dx is

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

91. The value of $\int_{0}^{1} 4 x^{3} \frac{d^{2}}{d x^{2}} {(1 - x^{2})}^{5} d x$ is