Integral Of Tan 2x Sec 2x

Decoding the Integral of tan(2x)sec²(2x): A Comprehensive Guide

The seemingly simple integral ∫tan(2x)sec²(2x) dx might initially appear daunting, but with a strategic approach and a solid understanding of trigonometric identities and substitution techniques, it unravels beautifully. This comprehensive guide will not only solve this integral but also delve into the underlying principles, providing you with a robust foundation for tackling similar problems.

Understanding the Problem: ∫tan(2x)sec²(2x) dx

Our goal is to find the antiderivative of the function f(x) = tan(2x)sec²(2x). This integral is a classic example showcasing the power of u-substitution, a fundamental technique in integration. The key lies in recognizing the relationship between the tangent and secant functions and cleverly choosing a substitution that simplifies the integrand.

The Power of U-Substitution

U-substitution, also known as the reverse chain rule, is a crucial tool for simplifying complex integrals. It involves substituting a portion of the integrand with a new variable 'u', making the integral easier to solve. After solving the simplified integral, we substitute back to express the result in terms of the original variable.

Choosing the Right Substitution

The strategic choice of 'u' is critical in u-substitution. In our case, observing the integrand tan(2x)sec²(2x), we notice that the derivative of tan(2x) is closely related to sec²(2x). This suggests a natural substitution:

Let u = tan(2x)

Now, we need to find the differential du:

Using the chain rule, the derivative of tan(2x) with respect to x is:

du/dx = 2sec²(2x)

Rearranging this equation gives us the differential we need:

du = 2sec²(2x) dx

Transforming the Integral

With our substitution, we can now rewrite the original integral in terms of 'u':

∫tan(2x)sec²(2x) dx = ∫ u * (1/2) du

Notice how the substitution has simplified the integrand significantly. The sec²(2x) term is now incorporated into the du, leaving us with a simple linear function of 'u' to integrate.

Solving the Simplified Integral

Integrating ∫ u * (1/2) du is straightforward:

∫ u * (1/2) du = (1/2) ∫ u du = (1/2) * (u²/2) + C

Where 'C' is the constant of integration. Simplifying this gives:

(1/4)u² + C

Back-Substitution and the Final Result

We have solved the integral in terms of 'u', but our original problem was expressed in terms of 'x'. To obtain the final solution, we need to substitute back our original substitution: u = tan(2x)

Substituting this back into our result, we get:

(1/4)(tan(2x))² + C

Therefore, the final solution to the integral ∫tan(2x)sec²(2x) dx is:

∫tan(2x)sec²(2x) dx = (1/4)tan²(2x) + C

Verifying the Solution: Differentiation

A crucial step in solving any integral problem is verifying the solution. We can do this by differentiating our result and confirming that we obtain the original integrand.

Let's differentiate (1/4)tan²(2x) + C with respect to x:

d/dx [(1/4)tan²(2x) + C] = (1/4) * 2tan(2x) * sec²(2x) * 2 (using the chain rule)

= tan(2x)sec²(2x)

This matches our original integrand, confirming the correctness of our solution.

Expanding the Understanding: Exploring Related Integrals

The technique demonstrated above forms a foundation for solving numerous related integrals. Let’s explore some examples:

1. ∫tan(ax)sec²(ax) dx (where 'a' is a constant)

This integral follows a similar pattern. We can use the substitution u = tan(ax), leading to du = a sec²(ax)dx. The solution becomes:

(1/(2a))tan²(ax) + C

2. ∫tan³(x)sec²(x) dx

This integral involves a higher power of tan(x). However, we can still employ a u-substitution. Let u = tan(x), and du = sec²(x)dx. Then the integral simplifies to:

∫u³ du = (1/4)u⁴ + C = (1/4)tan⁴(x) + C

3. Integrals involving other trigonometric functions

While the focus here was on tangent and secant, the same principles of substitution and manipulation can be extended to integrals involving other trigonometric functions such as sine, cosine, cosecant, and cotangent. The key is identifying appropriate substitutions that simplify the integrand to a form that can be readily integrated.

Advanced Techniques and Considerations

For more complex integrals, advanced techniques like integration by parts or trigonometric identities might be required in conjunction with u-substitution. These techniques offer broader tools for tackling intricate integration problems.

Conclusion: Mastering Integration Techniques

Mastering integration requires practice and a deep understanding of fundamental techniques like u-substitution. The integral of tan(2x)sec²(2x) serves as an excellent example of how a strategic substitution can significantly simplify a seemingly challenging problem. By understanding the underlying principles and practicing various examples, you can build a solid foundation in calculus and confidently approach even more complex integration problems. Remember the importance of verifying your solution through differentiation to ensure accuracy. The journey of mastering integration is a rewarding one, opening doors to a wider world of mathematical applications.