Rewriting and Substitution in an Indefinite Integral
Consider the integral \( I = \int \frac{x^2 – 2x}{\sqrt{x^2 – 2x – 3}}\,dx \) with \( x > 3 \). (a) Show that by completing the square, the numerator and the expression under the square root can be rewritten in terms of \( u = x – 1 \). (b) Using the substitution \( […]
Part a
Complete the square to rewrite the numerator and radicand. Show that \( x^2 – 2x = (x-1)^2 – 1 \) and \( x^2 – 2x – 3 = (x-1)^2 – 4 \). Show that setting \( u = x – 1 \) transforms the integral into \( \int \frac{u^2 – 1}{\sqrt{u^2 – 4}}\,du \).