Combining Definite Integrals

Combining integrals with equal or contiguous ranges

Combining integrals with either equal or contiguous ranges. For example:

$\int_{2}^{5} f(x)dx + \int_{2}^{5} g(x)dx = \int_{2}^{5}[f(x) + g(x)]dx$

$\int_{2}^{4} f(x)dx + \int_{4}^{3} f(x)dx = \int_{2}^{3} f(x)dx$

This section is really just a reiteration of what we have said earlier in these notes:

• You can integrate term by term or all at once in an integral:

$\int_{2}^{5} f(x) dx + \int_{2}^{5} g(x) dx = \int_{2}^{5}[f(x) + g(x)] dx$

• You can use the Fundamental Theorem of Calculus to simplify expressions:

$\int_{2}^{4} f(x) dx + \int_{4}^{3} f(x) dx = \int_{2}^{3} f(x) dx$

It is worth checking that you have an intuitive grasp of both of these and also that you have a formal understanding of why they work.

We will use the Fundamental Theorem of Calculus to unpack the second statement:

$\int_{2}^{4} f(x) dx + \int_{4}^{3} f(x) dx = \int_{2}^{3} f(x) dx + \int_{3}^{4} f(x) dx - \int_{3}^{4} f(x) dx$ $= \int_{2}^{3} f(x) dx$

Check you can see what we have done in the middle section.