 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/c/solution

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

t = x^{2} + 1

d t = 2 x d x

gives a logarithm. Thus

\int \frac{d x}{x ( x ^{2} + 1 )} = ln ∣ x ∣ - \frac{1}{2} ln (x^{2} + 1) + C, x \neq = 0.

The constant in the antiderivative may be chosen independently on the two intervals separated by

0

 New to topics? Read the docs here!