Solving The Indefinite Integral Of Sec²(θ)e^(6 Tan(θ))

Hey there, math enthusiasts! Today, we're diving headfirst into the fascinating world of indefinite integrals, tackling a particularly intriguing problem: ∫sec²(θ)e^(6 tan(θ)) dθ. Buckle up, because we're about to embark on a journey of trigonometric functions, exponential expressions, and the ever-elusive constant of integration, C. We will guide you through each step, ensuring you grasp the underlying concepts and techniques. So, grab your favorite beverage, find a cozy spot, and let's get started!

Setting the Stage: Understanding the Integral

Before we jump into the nitty-gritty, let's take a moment to appreciate the beauty of this integral. At first glance, it might seem a bit intimidating, but fear not! We're going to break it down piece by piece. Our main task here is to evaluate the indefinite integral $\int \sec ^2(\theta) e^{6 \tan (\theta)} d \theta$. Remember, an indefinite integral is the reverse process of differentiation. We're essentially trying to find a function whose derivative is sec²(θ)e^(6 tan(θ)). The presence of both trigonometric (sec²(θ) and tan(θ)) and exponential (e^(6 tan(θ))) functions suggests that a clever substitution might be our key to unlocking this integral. Keep in mind that when dealing with indefinite integrals, we always need to add the constant of integration, C, to account for all possible antiderivatives. This is a crucial step, as it acknowledges the fact that the derivative of a constant is zero.

Keywords: indefinite integral, trigonometric functions, exponential expressions, constant of integration, antiderivatives

The Power of Substitution: A U-Substitution Adventure

The heart of solving this integral lies in the elegant technique of u-substitution. This method allows us to simplify complex integrals by replacing a portion of the integrand with a single variable, u. The goal is to transform the integral into a more manageable form that we can readily integrate. Looking at our integral, ∫sec²(θ)e^(6 tan(θ)) dθ, a natural choice for u jumps out: u = 6 tan(θ). Why this choice? Well, the derivative of tan(θ) is sec²(θ), which conveniently appears in our integral. This is a strong indicator that this substitution will lead us down the right path. Now, let's find the derivative of u with respect to θ: du/dθ = 6sec²(θ). Rearranging this, we get du = 6sec²(θ) dθ, and further, sec²(θ) dθ = (1/6) du. See how smoothly this works? We've managed to express the sec²(θ) dθ part of our integral in terms of du. This is the magic of u-substitution in action! By carefully selecting our u, we've paved the way for a simpler integral.

Keywords: u-substitution, integrand, derivative, trigonometric functions, variable substitution

Transforming the Integral: A New Perspective

With our u-substitution defined (u = 6 tan(θ) and sec²(θ) dθ = (1/6) du), we can now rewrite our original integral in terms of u. This is where the integral starts to look significantly less daunting. Replacing 6 tan(θ) with u and sec²(θ) dθ with (1/6) du, we get: ∫sec²(θ)e^(6 tan(θ)) dθ = ∫e^u (1/6) du. Notice how the complex trigonometric and exponential expressions have been neatly transformed into a simple exponential function. The constant (1/6) can be pulled outside the integral, further simplifying our task: (1/6) ∫e^u du. Now, we're faced with a fundamental integral that most calculus students can recognize and solve with ease. This transformation highlights the power of u-substitution in simplifying complex integrals, making them solvable with basic integration rules. We've essentially changed the landscape of the problem, making it much easier to navigate.

Keywords: integral transformation, exponential function, constant factor, simplification, u-substitution

The Moment of Truth: Integrating e^u

Here comes the fun part – integrating e^u! This is a classic integral that has a beautiful and straightforward solution. The integral of e^u with respect to u is simply e^u. Remember, the exponential function e^x is its own derivative and its own integral (with respect to x, of course!). So, we have: (1/6) ∫e^u du = (1/6)e^u + C. Don't forget that crucial constant of integration, C! It's like the final flourish on a masterpiece, acknowledging the family of possible antiderivatives. We're almost there, guys! We've successfully integrated the transformed expression. However, we're not quite finished yet. Our final step is to bring everything back to the original variable, θ.

Keywords: exponential function, integral of e^u, constant of integration, antiderivatives, basic integration

The Grand Finale: Back to θ

We've come a long way, guys! We started with a complex integral, performed a clever u-substitution, simplified the expression, integrated it with ease, and now it's time for the final act: substituting back to our original variable, θ. Remember, we defined u = 6 tan(θ). So, to express our result in terms of θ, we simply replace u with 6 tan(θ): (1/6)e^u + C = (1/6)e^(6 tan(θ)) + C. And there you have it! We've successfully evaluated the indefinite integral. The final answer is (1/6)e^(6 tan(θ)) + C. This result beautifully combines exponential and trigonometric functions, showcasing the interconnectedness of mathematical concepts. Give yourselves a pat on the back for making it through this journey with me!

Keywords: back-substitution, original variable, trigonometric functions, exponential function, final answer

Expressing the Answer: Exact Form and Symbolic Notation

As the instructions specified, we've expressed our answer in exact form, using symbolic notation and fractions where needed. We have (1/6)e^(6 tan(θ)) + C, where (1/6) is a fraction, e^(6 tan(θ)) is in symbolic notation (using the exponential function), and C represents the arbitrary constant of integration. We've also diligently absorbed any other constants into C. This meticulous attention to detail ensures that our answer is not only mathematically correct but also adheres to the specific requirements of the problem. This is crucial in mathematical problem-solving, as precision and clarity are paramount.

Keywords: exact form, symbolic notation, fractions, arbitrary constant, mathematical precision

Wrapping Up: The Beauty of Integration

So, guys, we've conquered the integral ∫sec²(θ)e^(6 tan(θ)) dθ! We've seen how the power of u-substitution can transform seemingly complex integrals into manageable ones. We've revisited the fundamental integral of e^u and the crucial role of the constant of integration, C. More importantly, we've experienced the satisfaction of unraveling a mathematical puzzle, step by meticulous step. Remember, integration is more than just a mechanical process; it's a journey of discovery, a testament to the elegance and interconnectedness of mathematics. Keep practicing, keep exploring, and keep embracing the beauty of integration!

Keywords: integral evaluation, u-substitution technique, constant of integration, mathematical problem-solving, beauty of mathematics