Question

If ${\int }_{-1}^{4}f\left(x\right)dx=4$ and ${\int }_2^4[3-f(x\left)\right]dx=7$, then the value of ${\int }_{-1}^{2}f\left(x\right)dx$ is.

• A. $-2$
• B. $3$
• C. $5$
• D. $8$

Answer 1

The correct option is C $5$

${\int }_{-1}^{4}f\left(x\right)dx=4$ and ${\int }_2^4[3-f(x)dx=7]$

$F\left(4\right)-F(-1)=4$ $\to \left(1\right)$

and $3{\left|x\right|}_2^4-\left[F\right(4)-F(2\left)\right]=7$

$\left[F\right(4)-F(2\left)\right]=6-7=-1$

$\therefore$ $F\left(4\right)-F\left(2\right)=-1$$\to \left(2\right)$

${\int }_{-1}^{2}f\left(x\right)dx=F\left(2\right)-F(-1)$$\to \left(3\right)$

eqn.(1)-eqn.(2), we get $F\left(2\right)-F(-1)=5$

$\therefore$ ${\int }_{-1}^{2}f\left(x\right)dx=5$ (put value in eqn.(3) )