Question

Express the integrand as a sum of partial fractions and evaluate the integral.

What is the integral from 0 to 1 of x^3/ (x^2+8x+16) dx

Answer 1

First noting that \(x^2+8x+16=(x+4)^2\), you can use synthetic division to do some preliminary simplification. Below is the work for dividing \(x^3\) by \(x+4\):
\[\begin{array}{c|cccc}
-4&1&0&0&0\\
&&-4&16&-64\\
\hline
&1&-4&16&-64
\end{array}\]which translates to
\[\frac{x^3}{x+4}=x^2-4x+16-\frac{64}{x+4}\]Dividing the quotient by \(x+4\) (the non-remainder part is considered below), you have
\[\begin{array}{c|cccc}
-4&1&-4&16\\
&&-4&32\\
\hline
&1&-8&48
\end{array}\]which means
\[\frac{x^3}{(x+4)^2}=x-8+\frac{48}{x+4}-\frac{64}{(x+4)^2}\]