How to perform integration by substitution. (e.g. Find the integral of (2x)/((4+(3(x^2)))^2)) (10 marks)

We begin by looking at the integral by itself. The first thing we must do is evaluate what type of integration we should perform. It's always important to make sure that it's not a standard integral such as a function in the form f(x)= (g'(x))/(g(x)). Generally, integration by substitution is easier than integration by parts so I would always attempt a substitution if possible. So, how can we spot if substitution is possible? Some questions will tell you the substitution while others won't. Generally speaking if you have (f(x))^a you can substitute for 'u'. In our example, it's a little more difficult as our function has a quadratic squared on the denominator. However, when we differentiate the denominator it will cancel with the numerator, which will leave the function in terms of 'u'.For this substitution, we will substitute the quadratic on the denominator for 'u' (u=4+3(x^2)). As the original integral is with respect to 'x' we need to convert it to 'u'. We achieve this by differentiating our substitution with respect to ‘x’ which will give us du/dx=6x => dx=du/6x. We will now substitute our two parts into the original equation. ∫ ((2x)/(u^2))*(1/6x) du. Here’s where the substitution makes the question the easy, by the method of substitution, as the ‘2x’ and the ‘1/6x’ will cancel to make 1/3. This leaves us with ∫ 1/(3(u^2)) du. This can be written as (1/3) ∫(u^-2) du. What we have done is reduced a very complicated integral into something we can integrate in our heads. Therefore, after having completed the integral, our answer in terms of ‘u’ is -1/(3u) + c (never forget the ‘+ c’ for indefinite integrals). Leaving the answer in terms of ‘x’ just requires us to make the substitution from earlier (u=4+3(x^2)). Leaving the final answer as -1/(3(4+3(x^2))) + c.