HSC Arts 12th Board ExamMaharashtra State Board
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Solution for Prove that : ∫a−a f(x)dx=2∫a0f(x)dx , = 0, if f (x) is an odd function. - HSC Arts 12th Board Exam - Mathematics and Statistics

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Prove that : `int_-a^af(x)dx=2int_0^af(x)dx` , if f (x) is an even function.

                      = 0,                   if f (x) is an odd function.

Show that: `int_-a^af(x)dx=2int_0^af(x)dx` , if f (x) is an even function.

                      = 0,                   if f (x) is an odd function.

Solution 1

`int_-a^af(x)dx=int_-a^0f(x)dx+int_0^af(x)dx`

`int_a^af(x)dx=I+int_0^af(x)dx`

`Now I=int_-a^0f(x)dx`

Put x=-t

dx = - dt
When x = -a, t = a and when x = 0, t = 0

`I=int_a^0f(-t)(-dt)`

`=-int_a^0f(-t)dt`

`=int_0^af(-t)dt ...........[because int_a^bf(x)dx=-int_b^af(x)dx]`

`=int_0^af(-x)dt ...........[because int_a^bf(x)dx=-int_b^af(t)dx]`

Equation (i) becomes

`int_-a^af(x)dx=int_0^af(-x)dx+int_0^af(x)dx`

`=int_0^a[f(-x)+f(x)]dx.......(ii)`

case 1: If f(x) is an even function, then f(-x) = f(x).
Thus, equation (ii) becomes

`int_-a^af(x)dx=int_0^a[f(x)+f(x)]dx=2int_0^af(x)dx`

Case 2: If f(x) is an odd function, then f(-x) = -f(x).
Thus, equation (ii) becomes

`int_-a^af(x)dx=int_0^a[-f(x)+f(x)]dx=0`

Solution 2

We shall use the following results :

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Solution for question: Prove that : ∫a−a f(x)dx=2∫a0f(x)dx , = 0, if f (x) is an odd function. concept: Methods of Integration - Integration by Parts. For the courses HSC Arts, HSC Science (General) , HSC Science (Electronics), HSC Science (Computer Science)
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