I came across a question while evaluating the integral:

$$ \int_{0}^{\pi}\frac{\cos{t}}{1+9\sin^2{t}}\, dt $$

If you substitute $u=3\sin{t}$, you get:

$$ \int_{0}^{0}\frac{1}{3+3u^2}\, dt $$

Which evaluates to zero because(?) the bounds are both zero.

But then, can't you substitute any arbitrary expression to change both bounds to zero -- making the value zero? So in evaluating:

$$\int_{0}^{1}x\,dx$$

We could substitute $u = x^2-x$ or some trigonometric expression to change both bounds to zero. Clearly there is a misstep here, but which part of this substitution is invalid?

I came across a question while evaluating the integral:

$$ \int_{0}^{\pi}\frac{\cos{t}}{1+9\sin^2{t}}\, dt $$

If you substitute $u=3\sin{t}$, you get:

$$ \int_{0}^{0}\frac{1}{3+3u^2}\, du $$

Which evaluates to zero because(?) the bounds are both zero.

But then, can't you substitute any arbitrary expression to change both bounds to zero -- making the value zero? So in evaluating:

$$\int_{0}^{1}x\,dx$$

We could substitute $u = x^2-x$ or some trigonometric expression to change both bounds to zero. Clearly there is a misstep here, but which part of this substitution is invalid?