 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-1/17/b/ii/solution

Complete the square in the denominator and split the numerator into a multiple of its derivative plus a constant:

x^{2} + 4 x + 8 = (x + 2)^{2} + 4, x + 1 = \frac{1}{2} (2 x + 4) - 1.

Thus

\int \frac{x + 1}{x ^{2} + 4 x + 8} d x = \frac{1}{2} \int \frac{2 x + 4}{x ^{2} + 4 x + 8} d x - \int \frac{d x}{( x + 2 ) ^{2} + 4} = \frac{1}{2} ln (x^{2} + 4 x + 8) - \frac{1}{2} arctan \frac{x + 2}{2} + C .

The natural logarithm needs no absolute-value correction because its quadratic argument is strictly positive for every real

x

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