To determine

To write:

The given integral ∫-22f(x) dx+∫25fxdx-∫-2-1fxdx as a single integral in the form ∫abfxdx

Answer

∫-15f(x) dx

Explanation

1) Concept:

Use the properties of the integral:

i. ∫acfxdx+∫cbfxdx=∫abfxdx

ii. ∫bafxdx= - ∫abfxdx

2) Given:

∫-22f(x) dx+∫25fxdx-∫-2-1fxdx

3) Calculation:

By using the property of the integral (i),

∫-22f(x) dx+∫25fxdx-∫-2-1fxdx= ∫-25f(x) dx-∫-2-1fxdx

Now by using property (ii),

∫-25f(x) dx-∫-2-1fxdx= ∫-25f(x) dx+∫-1-2fxdx

So, again use property (i) to get,

∫-25f(x) dx+∫-1-2fxdx=∫-15f(x) dx

Therefore,∫-22f(x) dx+∫25fxdx-∫-2-1fxdx=∫-15f(x) dx

Conclusion:

∫-22f(x) dx+∫25fxdx-∫-2-1fxdx=∫-15f(x) dx