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