Quote from Essential Calculus: Early Transcendentals, by James Stewart:
If $f$ is continuous, then $$\int_{-\infty}^\infty f(x)dx=\lim_{t \to \infty}\int_{-t}^tf(x)dx$$
I thought this would be true if such a limit exists (aka if the area is convergent from $a \to \infty$ and from $-\infty \to a$), but the book answer-sheet marks it as false. Could anyone explain to me why it is false?
Thanks!
Take for example $f(x)=x$. Then $$ \int_{-t}^t f(x)\,dx=0, $$ since $f$ is an odd function, and hence the limit $\lim_{t\to\infty}\int_{-t}^t f(x)\,dx$ exists and it is equal to zero. However, the function $f(x)=x$ is NOT integrable on the whole of $\mathbb R$.
Another example is $$ \lim_{t\to\infty}\int_{-t}^t \frac{\sin x}{x}\,dx=\pi, $$ although $\dfrac{\sin x}{x}$ is also NOT integrable on the whole of $\mathbb R$.