 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-1/13/d/solution

Split the integral into

\int \frac{d x}{1 + x ^{2}} + \int \frac{x d x}{1 + x ^{2}} .

The first is the standard antiderivative

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

t = 1 + x^{2}

gives

\frac{1}{2} \int t^{- 1/2} d t = 1 + x^{2}

. Hence

ln (x + 1 + x^{2}) + 1 + x^{2} + C .

The logarithm's argument is positive for every real

x

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