Integrating f(x)=x(x+q)n and (px+q)n

Understanding Integration of f(x)=x(x+q)^n and (px+q)^n

Integration is an essential operation in calculus that serves as the reverse procedure of differentiation.
It plays a critical role in many areas of mathematics and its applications, including finding areas under curves or resolving differential equations.

When integrating the function f(x) = x(x+q)^n, it’s important to recognise it as a product of two simpler functions.
This situation calls for the application of the integration by parts method, a potent tool for functions that are products of two ‘parts’.
The formula used is ∫udv = uv - ∫vdu, where, in this case, u=x and dv/dx=(x+q)^n.

In contrast, when you’re tasked with integrating the function f(x) = (px+q)^n, you’re dealing not with a product, but a function of the form f(g(x)).
Here, you’ll need to use the substitution method or the chain rule in reverse, where ‘u’ substitution is often the first step.
Typically, you would let u=px+q, which simplifies the integral to ∫u^n du, which can then be integrated more straightforwardly.
The result will then need to be back-substituted to replace ‘u’ with the original terms in ‘x’.
Remember, when performing a ‘u’ substitution, you also need to express ‘dx’ in terms of ‘du’.

In both cases, you will eventually need to apply the power rule for integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C, where ‘C’ is the constant of integration.
This rule allows you to easily find the antiderivative for any power of ‘x’.

Be diligent while applying integration techniques and always double-check your work. A mistake early on can drastically affect your ultimate solution.
Using an eraser to correct errors is perfectly acceptable in calculus. Don’t be afraid to adjust your work as necessary.
Don’t forget to write your final answer neatly and clearly and always include the constant of integration, ‘+C’.

Practice is essential for mastering these techniques. Take advantage of practice problems to apply and reinforce these concepts.
Draw upon your understanding of differentiation, as many integration methods are effectively the reverse processes.