Integrating Exponential and Trigonometric Functions: A Step-by-Step Guide
In this comprehensive guide, we will explore the process of integrating a specific function involving exponential and trigonometric components. We will reflect on the method used to solve the integral of ( e^{2 sin x} sin^2 x cos x ,dx ). This example, often encountered in advanced calculus, serves as an excellent case study for understanding the application of different integration techniques, making it a valuable reference for students, researchers, and professionals working with complex mathematical integrals.
Problem Statement
The integral in question is:
( I int e^{2 sin x} sin^2 x cos x , dx )
Method 1: Substitution and Integration by Parts
The first approach involves using a substitution followed by integration by parts. Let's start by letting ( u sin x ), which implies ( du cos x , dx ). With this substitution, the integral becomes:
( int u^2 e^{2u} , du )
Next, we apply integration by parts. We choose:
( v u^2 ) and ( dv 2u , du ) ( w e^{2u} ) and ( dw 2 e^{2u} , du )Using the formula for integration by parts, ( int v , dw vw - int w , dv ), we get:
( int u^2 e^{2u} , du frac{1}{2} u^2 e^{2u} - int e^{2u} cdot 2u , du )
Applying integration by parts again to the remaining integral:
I frac{1}{2} u^2 e^{2u} - frac{1}{2} u e^{2u} cdot frac{1}{2} e^{2u} C
Substituting back ( u sin x ), we have:
( I frac{1}{4} e^{2 sin x} (1 - 2 sin x) C )
Method 2: Substitution and Series Expansion
Alternatively, the integral can be solved using a substitution followed by a series expansion. Let's start by setting:
( 2 sin x t ), which implies ( cos x , dx frac{1}{2} dt ). Then, the integral becomes:
( int frac{t^2}{4} cdot frac{1}{2} e^t , dt frac{1}{8} int t^2 e^t , dt )
For the second method, we can use a binomial series expansion for simplicity. Using the theorem:
( int e^x (f(x)f'(x)) , dx e^x f(x) C )
We can rewrite:
( int e^t t^2 , frac{dt}{2} frac{1}{2} e^t (t^2 - 2t 2) C )
Substituting back, we get:
( I frac{e^{2 sin x}}{4} (2 sin^2 x - sin x 1) C )
Conclusion
This extensive exploration of integrating a function involving exponential and trigonometric components showcases the versatility and power of multiple integration techniques. Whether through substitution, integration by parts, or series expansion, this case study demonstrates how to systematically solve such problems, ensuring a robust understanding for future problem-solving scenarios in advanced calculus.