solving definite integral with properties

Question

Could someone help me with this question?

Suppose that f(x) is a continuous and odd function such that $$\displaystyle\int^7_2f(x)\ d(x)=-3$$

Find $$\int^7_{-2}(1+f(x))\ dx$$

So far, I have used the properties of the integral to split the integral and evaluate it but I am stuck on how to change the sign of the 2 in the lower limit in the last integral from my work below. I know it has something to do with the integral being odd but I don't know what exactly.

My work so far: $$\int^7_{-2}(1+f(x))\ dx=\int^7_{-2}(1)\ dx \ +\ \int^7_{-2}f(x)\ dx=9\ +\ \int^7_{-2}f(x)$$

Answer 1

Since $f$ is odd,$$\int_{-2}^2f(x)\,\mathrm dx=0$$and therefore\begin{align}\int_{-2}^7\bigl(1+f(x)\bigr)\,\mathrm dx&=\int_{-2}^71\,\mathrm dx+\int_{-2}^2f(x)\,\mathrm dx+\int_2^7f(x)\,\mathrm dx\\&=9-3\\&=6.\end{align}