Find the Integral

Calculus Level 3

Given that ∫ 12 − 10 f ( x ) d x = 6 \displaystyle \int_{12}^{-10} f(x) \ dx = 6 , ∫ 100 − 10 f ( x ) d x = − 2 \displaystyle \int_{100}^{-10} f(x) \ dx = -2 , and ∫ 100 − 5 f ( x ) d x = 4 \displaystyle \int_{100}^{-5} f(x) \ dx = 4 , determine the value of ∫ − 5 12 f ( x ) d x \displaystyle \int_{-5}^{12} f(x) \ dx .


The answer is -12.

Aly Ahmed by Aly Ahmed , 34, Egypt

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3 solutions

Tom Engelsman
May 9, 2020

We require:

∫ − 5 12 f ( x ) d x = ∫ − 5 100 f ( x ) d x + ∫ 100 − 10 f ( x ) d x + ∫ − 10 12 f ( x ) d x = ( − 4 ) + ( − 2 ) + ( − 6 ) = − 12 . \int_{-5}^{12} f(x) dx = \int_{-5}^{100} f(x) dx + \int_{100}^{-10} f(x) dx + \int_{-10}^{12} f(x) dx = (-4) + (-2) + (-6) = \boxed{-12}.

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Chew-Seong Cheong
May 11, 2020

∫ − 5 12 f ( x ) d x = ∫ − 10 12 f ( x ) d x − ∫ − 10 − 5 f ( x ) d x = ∫ − 10 12 f ( x ) d x − ( ∫ − 10 100 f ( x ) d x − ∫ − 5 100 f ( x ) d x ) = − ∫ 12 − 10 f ( x ) d x + ∫ 100 − 10 f ( x ) d x − ∫ 100 − 5 f ( x ) d x = − 6 − 2 − 4 = − 12 \begin{aligned} \int_{-5}^{12} f(x)\ dx & = \int_{-10}^{12} f(x)\ dx - \int_{-10}^{-5} f(x)\ dx \\ & = \int_{-10}^{12} f(x)\ dx - \left(\int_{-10}^{100} f(x)\ dx - \int_{-5}^{100} f(x)\ dx \right) \\ & = - \int^{-10}_{12} f(x)\ dx + \int^{-10}_{100} f(x)\ dx - \int^{-5}_{100} f(x)\ dx \\ & = -6-2-4 = \boxed{-12} \end{aligned}

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Let ∫ f ( x ) d x = g ( x ) \displaystyle \int f(x)dx=g(x) . Then we have g ( − 10 ) − g ( 12 ) = 6 , g ( − 10 ) − g ( 100 ) = − 2 , g ( − 5 ) − g ( 100 ) = 4 g(-10)-g(12)=6, g(-10)-g(100)=-2, g(-5)-g(100)=4 .

Solving these we get g ( 12 ) − g ( − 5 ) = − 12 g(12)-g(-5)=-12 , that is, ∫ − 5 12 f ( x ) d x = − 12 \displaystyle \int_{-5}^{12} f(x)dx=\boxed {-12} .

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