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Question Number 55526 by alex26 last updated on 26/Feb/19

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

Commented by mr W last updated on 26/Feb/19

$$\underset{2}{\stackrel{4}{\int }}(3-f\left(x\right))dx=7$$$$\Rightarrow 3(4-2)-\underset{2}{\stackrel{4}{\int }}f\left(x\right)dx=7$$$$\Rightarrow \underset{2}{\stackrel{4}{\int }}f\left(x\right)dx=-1$$$$$$$$\underset{-1}{\stackrel{4}{\int }}f\left(x\right)dx=4$$$$\Rightarrow \underset{-1}{\stackrel{2}{\int }}f\left(x\right)dx+\underset{2}{\stackrel{4}{\int }}f\left(x\right)dx=4$$$$\Rightarrow \underset{-1}{\stackrel{2}{\int }}f\left(x\right)dx-1=4$$$$\Rightarrow \underset{-1}{\stackrel{2}{\int }}f\left(x\right)dx=5$$$$\Rightarrow -\underset{2}{\stackrel{-1}{\int }}f\left(x\right)dx=5$$$$\Rightarrow \underset{2}{\stackrel{-1}{\int }}f\left(x\right)dx=-5$$

Answered by tanmay.chaudhury50@gmail.com last updated on 26/Feb/19

$$\int f\left(x\right)dx=g\left(x\right)+c$$$${\int }_{-1}^{4}f\left(x\right)dx$$$$g\left(4\right)-g(-1)=4.....\left(1\right)eqn$$$${\int }_2^4\{3-f\left(x\right)\}dx$$$$\mid 3x-g\left(x\right){\mid }_2^4$$$$3\times 4-g\left(4\right)-3\times 2+g\left(2\right)$$$$6-g\left(4\right)+g\left(2\right)=7.....\left(2\right)eqn$$$$addeqn1andean2$$$$6+g\left(2\right)-g(-1)=11$$$$g\left(2\right)-g(-1)=5$$$$g(-1)-g\left(2\right)=-5$$$$so{\int }_2^{-1}f\left(x\right)dx=-5$$$$plscheck...$$$$$$

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