How do I evaluate this definite integral?
You are right that $\int_{-1}^0xf(x^2) \, dx$ is equal to $\frac{1}{2}\int_{1}^0f(x)\,dx$, and that is equal to $-\frac{1}{2}\int_{0}^1f(x)\, dx$.
So the integration from $-1$ to $0$ gives the value of $$ \int_{-1}^0xf(x^2)\,dx+3\int_{-1}^0f(x)\,dx = -\frac 12 \int_0^1 f(x)\, dx + 3\int_{-1}^0f(x)\,dx \, . $$ Now integrate the equation from $0$ to $1$, that gives the value of $$ \int_{0}^1xf(x^2)\,dx+3\int_{0}^1f(x)\,dx = \frac 12 \int_0^1 f(x) \,dx + 3\int_{0}^1f(x)\,dx \, . $$ Using both results you can now compute $\int_{-1}^0 f(x) \,dx$.