Evaluate \(\int_0^4\frac3{\sqrt{x}}\,dx\). Show how the lower endpoint is handled.

Evaluate \(\int_2^\infty\frac{12}{x^3}\,dx\), showing a limiting calculation.

A student writes \(\int_{-1}^1x^{-2}\,dx=[-x^{-1}]_{-1}^1=-2\). Explain the error and state whether the integral converges.

Which statement about \(\int_0^1x^{-1/2}\,dx\) is correct? Explain using a finite lower bound.

Evaluate \(\int_0^9 2/\sqrt{x}\,dx\), explaining the endpoint limit.

Evaluate \(\int_3^\infty18/x^4\,dx\), using a finite upper limit first.

A student subtracts endpoint values to claim \(\int_{-2}^{2}x^{-2}\,dx=-1\). State whether it converges and describe the one-sided contributions.