Before integrating \(\frac{x^3+2}{x+1}\), should polynomial division be used?

In polynomial division, what condition must the remainder \(R(x)\) satisfy when dividing by \(Q(x)\)?

Divide \(x^2+5x+6\) by \(x+2\).

Write \(\frac{x^2+1}{x-1}\) as a polynomial plus a proper fraction.

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

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

Divide \(2x^3+3x^2-x+4\) by \(x+2\).

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

Evaluate \(\int \frac{x^3+2x^2+x+5}{x+1}\,dx\).

Evaluate \(\int \frac{2x^3-x^2+4}{x^2+1}\,dx\).

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

A student writes \(\frac{x^3-1}{x-1}=x^2+1\). Explain the error and give the correct quotient.

Evaluate \(\int \frac{x^4-3x^2+2x+7}{x^2-1}\,dx\).

Evaluate \(\int \frac{3x^4+x^3+2x+1}{x^2+x}\,dx\).

Find \(a\) so that dividing \(x^3+ax+4\) by \(x-2\) leaves remainder \(0\), then write the quotient.

Find \(k\) so that \(\frac{x^3+kx+1}{x^2+1}\) has remainder \(1\) after division, then integrate the rational function.

For what values of \(m\) and \(n\) is \(x^3+mx+n\) exactly divisible by \(x^2-1\)?

A student divides \(x^4+2x^2+3\) by \(x^2+1\) and stops at \(x^2+2\) with remainder \(3\). Diagnose the error and give the correct decomposition.

Prove that if \(\deg P\ge \deg Q\) and \(Q\ne0\), polynomial division writes \(\frac{P(x)}{Q(x)}=S(x)+\frac{R(x)}{Q(x)}\) with \(\deg R<\deg Q\).

Evaluate \(\int \frac{x^5-x^4+x^2+1}{x^2+1}\,dx\), showing the division and the remaining standard integrals.