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`?
- `2 int_0^a f(x)\ dx`
- `2 int_0^a g(x)\ dx`
- `int_(−a)^a g(x)\ dx`
- `2int_0^a f(x) + g(x)\ dx`
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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`?
`A`
`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`
Without evaluating the integrals, which one of the following integrals is greater than zero?
(A) `int_(−pi/2)^(pi/2) x/(2 + cos x)\ dx`
(B) `int_(−pi)^pi x^3 sin x\ dx`
(C) `int_(−1)^1 (e^(−x^2) − 1)\ dx`
(D) `int_(−2)^2 tan^(−1)(x^3)\ dx`
`B`
`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`