Write the correct partial-fraction form for \(\frac{5x+1}{(x-2)(x+3)}\). Do not find the constants.

Write the correct partial-fraction form for \(\frac{x+4}{(x-1)^2}\). Do not find the constants.

Decompose \(\frac{7}{x(x+1)}\) into partial fractions.

Decompose \(\frac{3x+1}{(x-1)(x+2)}\).

Evaluate \(\int \frac{5}{x(x+5)}\,dx\).

Evaluate \(\int \frac{x+4}{(x+1)(x+3)}\,dx\).

Decompose \(\frac{2x+5}{(x-2)^2}\).

Write the correct partial-fraction form for \(\frac{4x^2+1}{(x+1)(x^2+4)}\), then identify the numerator type over \(x^2+4\).

Evaluate \(\int \frac{3x+7}{(x+1)(x+2)}\,dx\).

Evaluate \(\int \frac{x}{(x-1)(x+1)}\,dx\).

Evaluate \(\int \frac{x+3}{x^2(x+1)}\,dx\).

Explain why \(\frac{2x+1}{(x-1)^2(x+4)}\) needs three partial-fraction terms, then write the setup.

Evaluate \(\int \frac{2x^2+x+5}{(x+1)(x^2+4)}\,dx\).

Evaluate \(\int \frac{1}{(x-2)^2(x+1)}\,dx\).

Find \(a\) so that \(\frac{x+a}{(x-1)(x+2)}\) has equal partial-fraction coefficients, then decompose it.

For which \(k\) does \(\frac{kx+6}{(x-3)(x+1)}\) have no \(\frac{1}{x+1}\) term in its partial-fraction decomposition?

Find \(A,B,C\) if \(\frac{4x^2+px+q}{(x-1)(x+2)(x+3)}=\frac{A}{x-1}+\frac{B}{x+2}+\frac{C}{x+3}\) and \(A=1\), \(B=2\). Express \(C,p,q\).

A student decomposes \(\frac{x+1}{(x-2)^2}\) as \(\frac{A}{(x-2)^2}\) only. Explain why this is incomplete and find the correct decomposition.

Prove by coefficient counting why \(\frac{Ax+B}{x^2+4}\) is needed for an irreducible quadratic denominator, not just \(\frac{A}{x^2+4}\).

Evaluate \(\int \frac{3x^2+2x+11}{(x+1)(x^2+9)}\,dx\) and show the decomposition.