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  • Answer please! Which of the following equations will NOT be true?

Which of the following equations will NOT be true?

  • Option 1)

    \int_{-a}^{a}f(x)dx = \int_{-a}^{0}f(x)dx + \int_{0}^{a}f(x)dx

  • Option 2)

    \int_{a}^{c}f(x)dx = \int_{a}^{d}f(x)dx + \int_{d}^{c}f(x)dx \; ; where\; a<d<c

  • Option 3)

    \int_{a}^{c}f(x)dx = \int_{a}^{b}f(x)dx + \int_{b}^{c}f(x)dx \; ; where\; a<c<b

  • Option 4)

    All are true.

 

Answers (1)

As we have learnt,

 

Fundamental Properties of Definite integration -

If the function is continuous in (a, b ) then integration of a function a to b will be same as the sum of integrals of the same function from a to c and c to b.

\int_{b}^{a}f\left ( x \right )dx= \int_{a}^{c}f\left ( x \right )dx+\int_{c}^{b}f\left ( x \right )dx
 

- wherein

 

 

 

 It doesn't matter if b lies between a and c or not.

 


Option 1)

\int_{-a}^{a}f(x)dx = \int_{-a}^{0}f(x)dx + \int_{0}^{a}f(x)dx

Option 2)

\int_{a}^{c}f(x)dx = \int_{a}^{d}f(x)dx + \int_{d}^{c}f(x)dx \; ; where\; a<d<c

Option 3)

\int_{a}^{c}f(x)dx = \int_{a}^{b}f(x)dx + \int_{b}^{c}f(x)dx \; ; where\; a<c<b

Option 4)

All are true.

Posted by

Vakul

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