Question

If $\stackrel{4}{\underset{-1}{\int }}f\left(x\right)dx=4$ and $\stackrel{4}{\underset{2}{\int }}[3-f(x\left)\right]dx=7$, then $\stackrel{-1}{\underset{2}{\int }}f\left(x\right)dx=$

• A. $2$
• B. $-3$
• C. $-5$
• D. $1$

Answer 1

The correct option is A $2$

${\int }_{-1}^{4}f\left(x\right)dx=4$

${\int }_2^4[3-f(x\left)\right]dx$

changing limits from (2 to 4) to

(-1 to 4) - (-1 to 2)

$\therefore \Rightarrow {\int }_{-1}^4[3-f(x\left)\right]dx-{\int }_{-1}^2[3-f(x\left)\right]dx=7$

$\Rightarrow [{\int }_{-1}^{4}3dx-{\int }_{-1}^{4}f(x\left)dx\right]-$

$[{\int }_{-1}^{2}3dx-{\int }_{-1}^{2}f(x\left)dx\right]=7$

$\Rightarrow [3x{]}_{-1}^4-4-[3x{]}_{-1}^2-{\int }_{-1}^{2}f\left(x\right)dx=7$

${\int }_{-1}^{2}f\left(x\right)dx=7+4-3\times (4+1)+3\times (2+1)$

= 2.