Integration by substitution example

Question

If we decide to integrate from the limits $a$ to $b$, i.e. $\int^b_ax dx$. The function here doesn't matter, it's just an example. But if we then use the substitution $x=\frac{b-a}{2}t+\frac{b+a}{2}$ why is it that our new limits then will become $\int^1_{-1} f(x(t)) dt$. I don't see how $\frac{b-a}{2}a+\frac{b+a}{2}=-1$ or $\frac{b-a}{2}b+\frac{b+a}{2}=1$.

Specific image of example

Answer 1

It's because:

• $t=1\implies\frac{b-a}2t+\frac{b+a}2=b$;
• $t=-1\implies\frac{b-a}2t+\frac{b+a}2=a$.