If f is continuous function, then - Vidyadip

MCQ

If $f$ is continuous function, then

A
$\int_{ - 2}^2 {f(x)dx = \int_0^2 {[f(x) - f( - x)]dx} } $
B
$\int_{ - 3}^5 {2f(x)dx = \int_{ - 6}^{10} {f(x - 1)dx} } $
C
$\int_{ - 3}^5 {f(x)dx = \int_{ - 4}^4 {f(x - 1)dx} } $
✓
$\int_{ - 3}^5 {f(x)dx = \int_{ - 2}^6 {f(x - 1)dx} } $

✓

Answer

Correct option: D.

$\int_{ - 3}^5 {f(x)dx = \int_{ - 2}^6 {f(x - 1)dx} } $

d
(d) Since, $f$ is continuos function. Let $x = t - 1$

$\therefore $ $dx = dt$.

When $x = - 3 \to 5$, then $t = - 2 \to 6$

Therefore, $\int_{ - 3}^5 {f(x)dx} $

$ = \int_{ - 2}^6 {f(t - 1)dt = } \int_{ - 2}^6 {f(x - 1)dx} $.

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