 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2003/paper-1/4/solution

u = ln (1 + x^{2})

and

d v = (1 + 3 x^{2}) d x

v = x + x^{3} = x (1 + x^{2})

. The remaining integrand simplifies because

d u = 2 x d x / (1 + x^{2})

\int (1 + 3 x^{2}) ln (1 + x^{2}) d x = x (1 + x^{2}) ln (1 + x^{2}) - \int 2 x^{2} d x = x (1 + x^{2}) ln (1 + x^{2}) - \frac{2}{3} x^{3} + C .

The real formula is valid for every real number

x

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