Question

Which is the correct split for \[\int\frac{x+2}{2x^2+6x+5}\,dx\]?

Options

• \[\int\frac{4x+6}{2x^2+6x+5}\,dx+\int\frac{dx}{2x^2+6x+5}\]

• \[\frac14\int\frac{4x+6}{2x^2+6x+5}\,dx+\frac12\int\frac{dx}{2x^2+6x+5}\]

• \[\frac12\int\frac{4x+6}{2x^2+6x+5}\,dx+\frac14\int\frac{dx}{2x^2+6x+5}\]

• \[\frac14\int\frac{x+2}{4x+6}\,dx+\frac12\int\frac{dx}{2x^2+6x+5}\]

Answer 1

Use \[x+2=\frac14(4x+6)+\frac12\]. Dividing by the denominator gives the two integrals with coefficients \[\frac14\] and \[\frac12\].