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Question: If \(\int _ { - 1 } ^ { 4 } f ( x ) d x = 4\) and \(\int _ { 2 } ^ { 4 } ( 3 - f ( x ) ) d x = 7\...

If ∫−14f(x)dx=4\int _ { - 1 } ^ { 4 } f ( x ) d x = 4 and ∫24(3−f(x))dx=7\int _ { 2 } ^ { 4 } ( 3 - f ( x ) ) d x = 7 then the value of ∫2−1f(x)dx\int _ { 2 } ^ { - 1 } f ( x ) d x is

A

2

B

– 3

C

– 5

D

None of these

Answer

– 5

Explanation

Solution

We have ∫24(3−f(x))dx=7⇒6−∫24f(x)dx=7\int _ { 2 } ^ { 4 } ( 3 - f ( x ) ) d x = 7 \Rightarrow 6 - \int _ { 2 } ^ { 4 } f ( x ) d x = 7

⇒ ∫24f(x)dx=−1\int _ { 2 } ^ { 4 } f ( x ) d x = - 1.

Now, ∫2−1f(x)dx=−∫−12f(x)dx=−[∫−14f(x)dx+∫42f(x)dx]\int _ { 2 } ^ { - 1 } f ( x ) d x = - \int _ { - 1 } ^ { 2 } f ( x ) d x = - \left[ \int _ { - 1 } ^ { 4 } f ( x ) d x + \int _ { 4 } ^ { 2 } f ( x ) d x \right]

=−[∫−14f(x)dx−∫24f(x)dx]=−(4+1)=−5= - \left[ \int _ { - 1 } ^ { 4 } f ( x ) d x - \int _ { 2 } ^ { 4 } f ( x ) d x \right] = - ( 4 + 1 ) = - 5.