Question

If $f\left(x\right)$ is continuous in $(-1,4)$ and ${\int }_{-1}^{4}f\left(x\right)dx=4$ and ${\int }_2^4(3-f(x\left)\right)dx=7$, then ${\int }_{-1}^{2}f\left(x\right)dx=$

• A. $-2$
• B. $3$
• C. $5$
• D. $8$

Answer 1

Answer: (C)

$5$

We have,
${\int }_2^4(3-f(x\left)\right)dx=7$
$\Rightarrow 3{\int }_2^{4}1\cdot dx-{\int }_2^{4}f\left(x\right)dx=7$
$\Rightarrow 6-7={\int }_2^{4}f\left(x\right)dx$
$\Rightarrow {\int }_2^{4}f\left(x\right)dx=-1$
${\int }_{-1}^{4}f\left(x\right)dx=4$
$\Rightarrow {\int }_{-1}^{2}f\left(x\right)dx+{\int }_2^{4}f\left(x\right)dx=4$
$\Rightarrow {\int }_{-1}^{2}f\left(x\right)dx+(-1)=4$
$\Rightarrow {\int }_{-1}^{2}f\left(x\right)dx=5$