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Calculus, EXT2 C1 2017 HSC 7 MC

It is given that  `f(x)`  is a non-zero even function and  `g(x)`  is a non-zero odd function.

Which expression is equal to  `int_(−a)^a f(x) + g(x)\ dx`?

  1. `2 int_0^a f(x)\ dx`
  2. `2 int_0^a g(x)\ dx`
  3. `int_(−a)^a g(x)\ dx`
  4. `2int_0^a f(x) + g(x)\ dx`
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`A`

Show Worked Solution
`int_(−a)^a f(x) + g(x)\ dx` `= int_(−a)^a f(x)\ dx + int_(−a)^a g(x)\ dx`
  `= 2 int_0^a f(x)\ dx + 0`

`=> A`

Calculus, EXT2 C1 2012 HSC 10 MC

Without evaluating the integrals, which one of the following integrals is greater than zero?

  1. `int_(−pi/2)^(pi/2) x/(2 + cos x)\ dx`
  2. `int_(−pi)^pi x^3 sin x\ dx`
  3. `int_(−1)^1 (e^(−x^2) − 1)\ dx`
  4. `int_(−2)^2 tan^(−1)(x^3)\ dx` 
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`B`

Show Worked Solution

`text{Consider (A) and (D)}`

`f(x)=-f(-x)\ \ =>\ text(ODD functions where)`

`int_(−a)^a f(x)\ dx = 0`

`text{Consider (C)}`

`e^(−x^2)<1\ \ text(for all)\ x\ \ => e^(−x^2) − 1<0`

`:. text(Its graph is below the)\ x text(-axis and any integral)`

`text(will be negative)`

 

`text{Consider (B)}`

`text{(B)}\ text(is an even function where,)`

`x^3 sinx>=0\ \ text(for)\ \ \ -pi<=x<=pi`

`=>B`