For \(\frac{4x+1}{(x-2)(x+5)}=\frac{A}{x-2}+\frac{B}{x+5}\), what value of \(x\) finds \(A\) by cover-up?

State one condition under which the cover-up rule can be used directly.

Use cover-up to find \(A\) in \(\frac{3x+2}{(x-1)(x+4)}=\frac{A}{x-1}+\frac{B}{x+4}\).

Use cover-up to find \(B\) in \(\frac{3x+2}{(x-1)(x+4)}=\frac{A}{x-1}+\frac{B}{x+4}\).

Decompose \(\frac{5}{x(x+5)}\) using cover-up.

Decompose \(\frac{2x+7}{(x-3)(x+1)}\) using cover-up.

Use cover-up to decompose \(\frac{x^2+1}{(x-1)(x+1)(x+2)}\).

Use cover-up to evaluate \(\int \frac{6}{(x-1)(x+2)}\,dx\).

Use cover-up to evaluate \(\int \frac{4x+1}{(x-2)(x+3)}\,dx\).

Use cover-up to decompose \(\frac{2x^2-3}{x(x-1)(x+1)}\).

Explain why cover-up is not directly valid for \(\frac{x+1}{(x-2)^2}\), then decompose it by another method.

Use cover-up to evaluate \(\int \frac{x+5}{(x+1)(x+4)}\,dx\), and check the numerator after decomposing.

Decompose \(\frac{x^2+2x+4}{(x+1)(x+2)(x+3)}\) using cover-up.

Evaluate \(\int \frac{x^2+4}{x(x-2)(x+2)}\,dx\) using cover-up.

Find \(k\) so that the \(\frac{1}{x-2}\) term vanishes in the decomposition of \(\frac{kx+1}{(x-2)(x+3)}\).

For \(\frac{x+a}{(x-1)(x+1)}\), find \(a\) so the cover-up coefficients are opposites.

For \(\frac{x+a}{(x-2)(x+2)}\), find \(a\) so the two cover-up coefficients are equal.

A student uses cover-up on \(\frac{x+1}{(x-1)(x^2+4)}\) and writes \(\frac{A}{x-1}+\frac{B}{x^2+4}\). Diagnose the issue.

Justify the cover-up formula for \(A\) in \(\frac{P(x)}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b}\), where \(a\ne b\).

Use cover-up where valid to evaluate \(\int \frac{2x^2+x-1}{(x-2)(x+1)(x-1)}\,dx\), and explain the zero coefficient if it occurs.