Assume that $g(x)$ is an even function, and also that $\int_0^5 g(x) dx=3$ and $\int_5^7 g(x) dx=4$.

Question

Assume that g(x) is an even function, and also that $\int_0^5 g(x) dx=3$ and $\int_5^7 g(x) dx=4$.

Find $\int_0^{-5} g(x) dx$ and $\int_{-5}^7 g(x) dx$

My attempt

$$\int_0^{-5} g(x) dx=\int_0^5 g(x) dx + \int_{5}^{-5} g(x) dx$$

$$=3-\int_{-5}^5 g(x) dx$$ $$=3-2\int_{0}^5 g(x)$$ $$=3-2(3)=-3$$

$$\int_{-5}^{7} g(x) dx=\int_{-5}^5 g(x) dx + \int_{5}^7 g(x) dx$$ $$=2\int_{0}^5 g(x) + \int_{5}^7 g(x) dx$$ $$=2(3)+4=10$$

Can anyone please verify my answers

Answer 1

Correction:

$$\int_0^{-5} g(x) dx=\int_0^5 g(x) dx + \int_{\color{red}5}^{\color{red}{-5}} g(x) dx$$

$$=3-\int_{-5}^5 g(x) dx$$ $$=3-2\int_{0}^5 g(x)$$ $$=3-2(3)=-3$$

Or faster way:

$$\int_0^{-5} g(x) dx=-\int_{-5}^0 g(x) \, dx=-\int_0^5 g(x)\, dx=-3$$