How to Integrate (x)/(x^2+2x+4) - Step-by-Step Guide

Integration of (x)/(x^2+2x+4) - A Comprehensive Guide

This guide will walk you through the process of integrating the expression (x)/(x^2+2x+4) using the powerful technique of substitution.

Step 1: Substitution

Let's start by making a substitution to simplify the integral:

u = x^2 + 2x + 4

Step 2: Differentiation

Now, differentiate both sides of the equation with respect to x:

du/dx = 2x + 2

Step 3: Rearrangement

Rearrange the terms to isolate du:

du = (2x + 2) dx

Step 4: Substitution in the Integral

Substitute the values of u and du into the original integral:

Integral [(x)/(x^2+2x+4)] dx = (1/2) * Integral [du/u]

Step 5: Integration

Now we have a much simpler integral to solve. Integrate with respect to u:

Integral [(x)/(x^2+2x+4)] dx = (1/2) * ln|u| + C

Step 6: Back Substitution

Substitute the value of u back into the expression:

Integral [(x)/(x^2+2x+4)] dx = (1/2) * ln|x^2 + 2x + 4| + C

where C is the constant of integration.

Conclusion

Therefore, the solution to the integral of (x)/(x^2+2x+4) is:

Integral [(x)/(x^2+2x+4)] dx = (1/2) * ln|x^2 + 2x + 4| + C

This step-by-step guide has demonstrated how the substitution method can effectively simplify complex integrals and provide a clear path to finding the solution.