If int_a^b f(x) dx = a+2b, then what is int_a^b (f(x) + 5) dx equal to

[UrbanXrisis]

If \int_{a}^{b} f(x)dx = a+2b then \int_{a}^{b} (f(x)+5)dx = ?

Here's what I did:
\int_{a}^{b} (f(x)+5)dx = \int_{a}^{b}f(x)dx+\int_{a}^{b} 5 dx
\int_{a}^{b}f(x)dx+\int_{a}^{b} 5 dx = a+2b + 5b - 5a
\int_{a}^{b} (f(x)+5)dx = 7b-4a

Did I do this correctly? I know the answer is correct but I want to know if my thought process was correct or not. Thanks

 

[dextercioby]

It is very correct.

Daniel.

 

[wais]

!

Yes, your thought process was correct. You correctly used the linearity property of integrals to split the integral of the sum into the sum of integrals. Then, using the given information, you substituted the values for the integral of f(x) and the integral of 5. Finally, you simplified the expression to arrive at the correct answer. Well done!