HSC Science (General) 12th Board ExamMaharashtra State Board
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# Prove that int_a^bf(x)dx=f(a+b-x)dx - HSC Science (General) 12th Board Exam - Mathematics and Statistics

ConceptMethods of Integration - Integration by Substitution

#### Question

Prove that int_a^bf(x)dx=f(a+b-x)dx. Hence evaluate : int_a^bf(x)/(f(x)+f(a-b-x))dx

#### Solution

"Let "I = int_a^bf(x)dx

Put x= a + b - t

∴ dx = -dt

When x = a, t = b and when x = b, t = a

therefore I = int_b^af(a+b-t)(-dt)

therefore I = -int_b^af(a+b-t)dt

therefore I = int_a^bf(a+b-t)dt ... [because int_a^bf(x)dx=-int_b^af(x)dx]

therefore int_a^bf(x)dx = int_a^bf(a+b-x)dx ... [because int_a^bf(x)dx= int_a^bf(t)dt]

"Let "I = int_a^b(f(x))/(f(x)+f(a+b-x))dx ... (i)

therefore I = int_a^b(f(a+b-x))/(f(a+b-x)+f(a+b-(a+b-x)))dx

therefore I = int_a^b(f(a+b-x))/(f(a+b-x)+f(x))dx ... (ii)

Adding (i) and (ii) we get

2I = int_a^b(f(x)+f(a+b-x))/(f(x)+f(a+b-x))dx

therefore 2I = int_a^b1dx

therefore 2I = [x]_a^b

therefore I = (b-a)/2

Is there an error in this question or solution?

#### APPEARS IN

2013-2014 (October) (with solutions)
Question 6.2.3 | 4.00 marks

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Solution Prove that int_a^bf(x)dx=f(a+b-x)dx Concept: Methods of Integration - Integration by Substitution.
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