Properties of Definite Integrals MCQ Quiz in मराठी - Objective Question with Answer for Properties of Definite Integrals - मोफत PDF डाउनलोड करा
Last updated on Mar 18, 2025
Latest Properties of Definite Integrals MCQ Objective Questions
Properties of Definite Integrals Question 1:
समजा, \( f \) आणि \( g \) हे \( [ 0 , a] \) वरील सतत फल आहेत की, \( f ( x ) = f ( a - x ) \) आणि \( g ( x ) + g ( a - x ) = 4 \), तर \( \int _ { 0 } ^ { a } f ( x ) g ( x ) d x \) बरोबर किती:-
Answer (Detailed Solution Below)
Properties of Definite Integrals Question 1 Detailed Solution
\( I = \int _{ { 0 } }^{ { { a } } }{ f } ({ x }){ g }({ x }){ dx } \)
\( I = \int _{ { 0 } }^{ { { a } } }{ f } ({ a }-{ x }){ g }({ a }-{ x }){ d }{ x } \)
\( I = \int _{ { 0 } }^{ { { a } } }{ f } ({ x })(4-{ g }({ x })){ d }{ x } \)
\( I = 4 \int _{ { 0 } }^{ { { a } } }{ f } ({ x }){ d }{ x }-{ I } \)
\( \Rightarrow { I }=2 \int _{ { 0 } }^{ { { a } } }{ f } ({ x }){ d }{ x } \)
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Properties of Definite Integrals Question 2:
समजा, \( f \) आणि \( g \) हे \( [ 0 , a] \) वरील सतत फल आहेत की, \( f ( x ) = f ( a - x ) \) आणि \( g ( x ) + g ( a - x ) = 4 \), तर \( \int _ { 0 } ^ { a } f ( x ) g ( x ) d x \) बरोबर किती:-
Answer (Detailed Solution Below)
Properties of Definite Integrals Question 2 Detailed Solution
\( I = \int _{ { 0 } }^{ { { a } } }{ f } ({ x }){ g }({ x }){ dx } \)
\( I = \int _{ { 0 } }^{ { { a } } }{ f } ({ a }-{ x }){ g }({ a }-{ x }){ d }{ x } \)
\( I = \int _{ { 0 } }^{ { { a } } }{ f } ({ x })(4-{ g }({ x })){ d }{ x } \)
\( I = 4 \int _{ { 0 } }^{ { { a } } }{ f } ({ x }){ d }{ x }-{ I } \)
\( \Rightarrow { I }=2 \int _{ { 0 } }^{ { { a } } }{ f } ({ x }){ d }{ x } \)