Integration of Complex Functions: A Detailed Guide

Introduction

Integration is a fundamental concept in calculus with a wide range of applications in physics, engineering, and mathematics. This article focuses on the integration of a specific function: ∫3x - 1cos2xdx This function combines simple linear and trigonometric components, presenting unique challenges in integration. We'll explore various methods and techniques to solve this problem, including integration by parts, substitution, and breaking down the integral.

Method of Integration

Step 1: Rewriting the Integral

First, we rewrite the given function in a more suitable form for integration. Recall that (frac{1}{cos^2 x} sec^2 x). Therefore, the integral can be expressed as:

∫3x - 1cos2xdx ∫ (3x - 1)sec^2 x dx

Step 2: Integration by Parts

Next, we apply integration by parts. Integration by parts is a technique derived from the product rule for differentiation, allowing us to transform the integral into a simpler form.

u 3x - 1 dv sec^2 x dx du 3 dx v tan x

Using the integration by parts formula (∫ u dv uv - ∫ v du), we have:

∫(3x - 1) sec^2 x dx (3x - 1)tan x - ∫ 3tan x dx

Step 3: Solving the Integral of tan x

The next step is to integrate (3tan x). We know that:

∫ 3tan x dx 3 ∫ tan x dx 3ln |sec x| C

Step 4: Combining All Parts

Substituting the result back into the equation, we get the final solution:

∫3x - 1cos2xdx (3x - 1)tan x - 3ln |sec x| C

Conclusion

Integration of complex functions like (frac{3x - 1}{cos^2 x}) involves a combination of algebraic manipulation and integration techniques. Understanding and mastering these methods is crucial for solving more intricate problems in calculus.

Feel free to explore more complex functions and their integrals using similar techniques. Whether you are a beginner or an advanced calculus student, practice is the key to getting better at integration.