Part b
Using your partial fractions result from part (a), evaluate \(\int \frac{2x+3}{x^2+3x+2} \; dx\).
Integration of Rational Function with Linear Denominator
Evaluate the integral \(\int \frac{1}{x+4}\,dx\).
Partial Fractions Integration Yields ln and arctan Terms
Evaluate the integral \(\int \frac{2x+3}{(x-1)(x^2+4)}\,dx\) by decomposing it into partial fractions and then integrating to obtain terms involving \(\ln|x-1|\) and \(\arctan(\frac{x}{2})\).
Part b
Using the partial fraction decomposition from part (a), evaluate the indefinite integral \(\int \frac{2x+5}{(x+1)(x-2)}\,dx\). Then, express the definite integral \(\int_{p}^{q} \frac{2x+5}{(x+1)(x-2)}\,dx\) using the antiderivative.
Integration via Partial Fractions and Substitution
Evaluate the indefinite integral \(\int \frac{x^2+5x+6}{(x+1)(x^2+4)}\,dx\). Begin by decomposing the rational function into partial fractions and then apply an appropriate substitution where necessary to complete the integration.
Integration Domain Considerations for ln|x|
Consider the indefinite integral \(\int \frac{1}{x\ln|x|}\, dx\). Determine the domain of the integrand and discuss the implications for its integration.