Evaluate the integral \int \frac{x}{x tan(x) + 1} dx using integration by substitution, hence evaluate \int \frac{x}{x cot(x) - 1} dx.
(STEP I 2017, Q1i)For the first part, the hint u = x sin(x) + cos(x) is given. It can be seen that this is the denominator once the fraction is multiplied by cos(x) / cos(x). The answer is ln(x sin(x) + cos(x)) + c.For the second part, there is no hint given, but we can see it must be similar to the previous part. Multiplying the fraction by sin(x) / sin(x) makes the solution clear, to use another substitution, this time u = x cos(x) - sin(x). This will again give a similar answer of - ln(x cos(x) - sin(x)) + c.
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