The integration of e to the power of sin x, or ∫e^sinx dx, is a fascinating topic that combines exponential functions, trigonometric functions, and advanced calculus techniques. Exponential function, specifically e^sinx, exhibits unique behavior when integrated, contrasting with simpler forms. Trigonometric functions, such as sin x, introduce periodicity and oscillation, complicating the integration process. Advanced calculus methods, including series expansions and special functions, provide ways to approach this integral.
Ever felt like math problems are just messing with you? Well, get ready for a rollercoaster, because today we’re diving into a classic calculus conundrum: the integral of exsin(x). Yes, that’s right, we’re tackling the ∫ exsin(x) dx beast!
But don’t worry, I promise it’s not as scary as it looks (okay, maybe a little scary, but in a fun way!). This integral isn’t just some random exercise your professor threw at you; it’s a shining example of how cool integration can be. It pops up in all sorts of places, from electrical engineering to quantum mechanics, showing that math really is everywhere.
Think of exsin(x) as the Batman and Robin of functions – exponential growth and sinusoidal oscillation teaming up. Together, they create a function that’s both dynamically changing and rhythmic, making its integral a fascinating puzzle to solve.
Our mission, should you choose to accept it, is to demystify this integral. We will solve the integral ∫ exsin(x) dx with Integration by Parts, a clever technique that’s like the Swiss Army knife of calculus. Get ready to see some mathematical wizardry!
Prerequisites: Essential Building Blocks for Your Integration Adventure!
Before we bravely venture into the land of integrating exsin(x), let’s make sure our backpacks are packed with all the essential goodies. Think of this as our pre-flight checklist for calculus success! We need to be totally comfortable with a few key players: the exponential function, the sine function, their mischievous derivatives, and the mysterious world of indefinite integrals. So, let’s get this show on the road!
Exponential Function (ex) – The Ever-Growing Wonder
Ah, ex, that incredibly important exponential function! It’s basically nature’s favorite way to grow things. Imagine a population of bunnies that doubles at an insane rate – that’s ex in action! Mathematically speaking, ex is a function where the rate of change is equal to its current value. Crazy, right? Its derivative and integral are both ex, making it a unique and super handy function in calculus. I like to call it the “easy one” because the derivative never changes! To truly appreciate its awesomeness, picture a graph of ex: starting close to zero on the left and shooting sky-high as you move to the right. Now, that’s what I call exponential growth!
Sine Function (sin(x)) – The Wavy Wonder
Next up, we have sin(x)! Think of it as a calm wave on a peaceful beach or the smooth swaying of a windy day! It’s a fundamental trigonometric function that oscillates smoothly between -1 and 1. We’ll want to remember a few things about this sine function as it is periodic. It repeats its behavior in a cycle. The derivative of sin(x) is cos(x) and the integral is -cos(x). Keep these in mind as we go to integration! If you sketch a graph of sin(x), you’ll see those lovely, repeating waves I was telling you about that make it so darn useful for modeling all sorts of things.
Derivatives of ex and sin(x) – Knowing Your Transformations
Okay, now that we’ve met our functions, let’s talk transformations! Specifically, their derivatives. Remember that the derivative of ex is, well, ex! I told you it never changes! It’s like the chameleon of functions – always itself. And the derivative of sin(x) is cos(x). So Why is knowing this important? Well, when we do Integration by Parts, we’re going to be working with the derivatives of u and dv so knowing these by heart is going to make your integration by parts journey easier. Trust me on this.
Indefinite Integrals and the Constant of Integration – The Missing Piece of the Puzzle
Finally, let’s briefly review indefinite integrals. An indefinite integral is basically the antiderivative of a function. It’s like asking: “What function, when differentiated, gives me this?”. The notation for an indefinite integral looks like this: ∫ f(x) dx. The function “f(x)” is the function we’re trying to integrate and dx is the indication to do an integral with respect to “x”. This notation is important to know. Also, when finding indefinite integrals, we must never forget our trusty friend, the constant of integration, C! Why? Because the derivative of any constant is always zero, so when we reverse the process, we need to account for all possible constants that could have been there. C is essential. Leaving it out is like baking a cake and forgetting the flour.
With these prerequisites firmly in our grasp, we’re ready to dive headfirst into the wonderful world of integrating exsin(x). Let’s go!
The Integration by Parts Technique: A Step-by-Step Guide
Alright, let’s get down to the nitty-gritty of integration by parts! Think of this as the calculus equivalent of a cooking technique—sometimes you need to sauté, sometimes you need to bake, and sometimes you need to integrate by parts. This technique is especially handy when you have a product of two different types of functions (like our e^x and sin(x) combo) that you’re trying to integrate. Trust me, it’s a lifesaver!
General Formula
The magic incantation, I mean formula, for integration by parts is:
∫ u dv = uv – ∫ v du
Easy peasy, right? Okay, maybe not at first glance. Let’s break it down.
u: This is a function you’ll choose from your original integral – think of it as the easy-to-differentiate part.dv: This is the other function in your integral, which you’ll integrate.du: This is the derivative of your chosenu.v: This is the integral of your chosendv.
The goal here is to turn a difficult integral (∫ u dv) into a potentially easier one (∫ v du). It’s like trading a grumpy cat for a purring kitten—hopefully, it makes your life easier!
Choosing ‘u’ and ‘dv’
Now, the real trick is figuring out what to choose as u and dv. This is where a little strategy comes in handy. One popular mnemonic is LIATE or ILATE:
- L: Logarithmic functions (like ln(x))
- I: Inverse trigonometric functions (like arctan(x))
- A: Algebraic functions (like x2, x3)
- T: Trigonometric functions (like sin(x), cos(x))
- E: Exponential functions (like ex)
The idea is to choose u as the function that comes earlier in this list. Why? Because usually, differentiating them makes things simpler. For instance, differentiating a logarithmic function can turn it into a rational function, which is often easier to deal with.
A good choice here is paramount. If you pick the wrong u and dv, you might end up with a more complicated integral than you started with – kind of like trying to untangle headphones with your eyes closed!
Applying Integration by Parts to ∫ exsin(x) dx (First Application)
Let’s put this into practice with our integral, ∫ e^x sin(x) dx.
Following the LIATE rule, we see that sin(x) (Trigonometric) comes before e^x (Exponential). So, we’ll choose:
u = sin(x)dv = e^x dx
Now, let’s find du and v:
du = cos(x) dx(The derivative of sin(x) is cos(x))v = e^x(The integral of e^x is e^x)
Plugging these into our integration by parts formula, we get:
∫ e^x sin(x) dx = e^x sin(x) – ∫ e^x cos(x) dx
Boom! We’ve made some progress. But, uh oh, we still have an integral to solve: ∫ e^x cos(x) dx. Looks like we’re not quite out of the woods yet.
Applying Integration by Parts (Second Application)
Time for round two! We need to apply integration by parts again, this time to the integral ∫ e^x cos(x) dx.
Using the same LIATE logic:
u = cos(x)dv = e^x dx
Now, let’s find du and v (again!):
du = -sin(x) dx(The derivative of cos(x) is -sin(x))v = e^x(The integral of e^x is still e^x—thank goodness!)
Plugging these into the integration by parts formula, we get:
∫ e^x cos(x) dx = e^x cos(x) – ∫ e^x (-sin(x)) dx
Simplifying, we have:
∫ e^x cos(x) dx = e^x cos(x) + ∫ e^x sin(x) dx
And that’s it!
Recognizing the Cycle: “Wait, Haven’t We Done This Before?”
Okay, so you’ve bravely battled through a round (or two!) of integration by parts. You’re feeling pretty good about yourself, meticulously applying the formula, choosing your u and dv with the wisdom of Solomon…and then BAM! You realize something fishy is going on. The integral you’re staring at now? It looks suspiciously like the one you started with! Don’t panic; this isn’t a glitch in the Matrix. This is the cyclic nature of integration, and it’s totally normal (especially with functions like exsin(x) that just love to mess with you).
Substituting Back into the Original Equation: Untangling the Mess
Here’s where the magic happens. Remember that massive equation you painstakingly created, filled with es and sins and coss? It’s time to put it to work. You should have something resembling this:
∫ exsin(x) dx = exsin(x) – [excos(x) + ∫ exsin(x) dx]
Don’t be intimidated! Think of it like a puzzle. Let’s simplify things a bit:
∫ exsin(x) dx = exsin(x) – excos(x) – ∫ exsin(x) dx
See that ∫ exsin(x) dx on both sides? That’s our original integral, playing peek-a-boo.
Solving for the Integral: The Grand Finale
Now, for the pièce de résistance! We’re going to treat that ∫ exsin(x) dx like an x in algebra and solve for it. Add ∫ exsin(x) dx to both sides of the equation:
2∫ exsin(x) dx = exsin(x) – excos(x)
Almost there! Now, just divide both sides by 2 to isolate our integral:
∫ exsin(x) dx = (1/2) [exsin(x) – excos(x)]
And the grand finale: don’t forget the + C, the constant of integration! Because, you know, we always have to remember the little things.
∫ exsin(x) dx = (1/2) [exsin(x) – excos(x)] + C
There you have it! You’ve successfully navigated the cyclic nature of integration and solved for the integral of exsin(x). Give yourself a pat on the back – you’ve earned it!
Verification and Final Result: Ensuring Accuracy
Alright, we’ve wrestled with this integral, looped around it like a dog chasing its tail, and finally pinned it down. But before we declare victory and pop the champagne (or sparkling cider, whatever floats your boat), let’s make absolutely sure we got it right. How do we do that? By good ol’ differentiation!
Differentiating the Result: Did We Really Do It?
Remember, differentiation is basically integration’s grumpy older sibling. It undoes what integration does. So, if we take the derivative of our answer, we should end up right back where we started: e<sup>x</sup>sin(x). Let’s see if that actually happens.
We need to take the derivative of (1/2) * [e<sup>x</sup>sin(x) - e<sup>x</sup>cos(x)] + C. Now, I won’t bore you with all the nitty-gritty steps (you can do that on your own, and it would be great if you did!), but trust me (or, better yet, verify yourself) that after applying the product rule and a little bit of algebraic magic, you’ll find that:
d/dx [(1/2) * (e<sup>x</sup>sin(x) - e<sup>x</sup>cos(x)) + C] = e<sup>x</sup>sin(x)
Boom! Just like we wanted. The constant of integration C vanished into thin air (as it should, derivatives of constants are zero) and we are left with the original function. This confirms that our integration was correct. Give yourself a pat on the back; you’ve earned it!
The Grand Finale: Behold, The Final Answer!
After all that hard work, it’s time for the big reveal, drumroll please…
∫ e<sup>x</sup>sin(x) dx = (1/2) * ex[sin(x) – cos(x)] + C
There you have it, folks! That’s the integral of e<sup>x</sup>sin(x). It’s not the prettiest thing in the world, but it’s our ugly duckling, and we should be proud of it.
Recap: A Quick Journey Back
Let’s quickly recap the highlights of our adventure:
- We bravely faced the integral of
e<sup>x</sup>sin(x) dx. - We wielded the mighty integration by parts technique…twice!
- We danced the cyclic integration jig, where the integral kept reappearing.
- We solved for the integral by treating it like an algebraic variable
- Finally, we verified our result using differentiation, ensuring that we didn’t make any silly mistakes.
You’ve conquered a challenging integral today! Congratulations, you are awesome!
Further Exploration: Level Up Your Integration Game!
So, you’ve conquered the integral of exsin(x) – give yourself a pat on the back! But hold on, the calculus adventure doesn’t end here. Think of this integral as your level one boss. Now it’s time to explore the vast landscape of integration and unlock some serious skills. Get ready to continue your journey into other calculus concepts!
Trigonometric Identities: Your Secret Weapons
Remember those trigonometric identities you learned (or maybe tried to learn) in trigonometry? Well, dust them off, because they’re about to become your best friends in more complex integrals! Integrals involving functions like sin2(x), cos3(x), or even more exotic combinations can often be tamed with the right trigonometric identity. They’re like secret cheat codes for calculus! Instead of brute-forcing your way through, a clever identity can transform a seemingly impossible integral into something much more manageable. So, don’t underestimate the power of a well-placed sin2(x) + cos2(x) = 1!
Beyond Integration by Parts: A Toolkit of Techniques
While integration by parts is a powerful tool (as we’ve seen!), it’s not the only trick up calculus’ sleeve. There’s a whole arsenal of integration techniques waiting to be mastered, each with its own strengths and weaknesses.
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Substitution (u-substitution): This technique is all about simplifying the integral by replacing a complicated expression with a single variable, ‘u’. It’s like giving your integral a makeover!
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Trigonometric Substitution: When you encounter integrals involving square roots of quadratic expressions (think √(a2 – x2)), trigonometric substitution can be a lifesaver. It involves replacing ‘x’ with a trigonometric function, which cleverly eliminates the square root.
Definite Integrals: Putting Limits on Your Calculus
Up until now, we’ve been dealing with indefinite integrals, which give us a family of functions that differ by a constant (the ever-present “+ C”). But what if we want to find the exact area under a curve between two specific points? That’s where definite integrals come in! A definite integral has limits of integration (a and b), which define the interval over which we’re calculating the area. Evaluating a definite integral involves finding the indefinite integral first, then plugging in the limits of integration and subtracting the results. Suddenly your function turns into a specific number representing area, distance or any accumulation function.
How does one methodically approach the integration of e^x sin(x)?
The integration of e^x sin(x) requires integration by parts because it involves a product of exponential and trigonometric functions. Integration by parts is a calculus technique that simplifies integrals of products. The formula is ∫u dv = uv – ∫v du, where u and dv are parts of the original integrand. We choose u as sin(x) because its derivative simplifies with each iteration. Consequently, dv is e^x dx.
The derivative of sin(x) is cos(x), making du equal to cos(x) dx. The integral of e^x dx is e^x, so v equals e^x. Applying the integration by parts formula yields ∫e^x sin(x) dx = e^x sin(x) – ∫e^x cos(x) dx. We apply integration by parts again to the new integral ∫e^x cos(x) dx.
In the second iteration, u is cos(x) and dv is e^x dx. The derivative of cos(x) is -sin(x), thus du is -sin(x) dx. The integral of e^x dx remains e^x, so v is e^x. Applying integration by parts results in ∫e^x cos(x) dx = e^x cos(x) + ∫e^x sin(x) dx. Substituting this back into the first equation gives ∫e^x sin(x) dx = e^x sin(x) – (e^x cos(x) + ∫e^x sin(x) dx).
Simplifying the equation involves algebraic manipulation to isolate the original integral. We add ∫e^x sin(x) dx to both sides of the equation, resulting in 2∫e^x sin(x) dx = e^x sin(x) – e^x cos(x). Dividing both sides by 2 isolates the integral, giving ∫e^x sin(x) dx = (e^x sin(x) – e^x cos(x))/2. Adding the constant of integration completes the process, resulting in ∫e^x sin(x) dx = (e^x sin(x) – e^x cos(x))/2 + C.
What is the role of complex numbers in evaluating the integral of e^(ax) sin(bx)?
Complex numbers offer a powerful technique for evaluating integrals of the form ∫e^(ax) sin(bx) dx because they unify exponential and trigonometric functions. Euler’s formula states e^(ix) = cos(x) + i sin(x), linking complex exponentials to trigonometric functions. The function sin(bx) is the imaginary part of e^(ibx). Therefore, e^(ax)sin(bx) is the imaginary part of e^(ax)e^(ibx) = e^((a+ib)x).
Integrating e^((a+ib)x) is straightforward using standard exponential integration. The integral of e^((a+ib)x) is e^((a+ib)x) / (a+ib) + C, where C is the constant of integration. To find ∫e^(ax) sin(bx) dx, we take the imaginary part of the result. The term e^((a+ib)x) can be rewritten as e^(ax) (cos(bx) + i sin(bx) using Euler’s formula.
Dividing by (a+ib) requires multiplying the numerator and denominator by the conjugate (a-ib). Thus, [ \frac{e^{(a+ib)x}}{a+ib} = \frac{e^{ax}(\cos(bx) + i\sin(bx))}{a+ib} \cdot \frac{a-ib}{a-ib} ]. Expanding and simplifying separates the real and imaginary parts.
The imaginary part represents the solution to ∫e^(ax) sin(bx) dx. After simplification, the imaginary part becomes [ \frac{e^{ax}(a\sin(bx) – b\cos(bx))}{a^2 + b^2} ]. Adding the constant of integration completes the solution, resulting in ∫e^(ax) sin(bx) dx = e^(ax) (a sin(bx) – b cos(bx)) / (a^2 + b^2) + C.
How does tabular integration simplify integrating e^x sin(x)?
Tabular integration, also known as the “Tic-Tac-Toe” method, simplifies the integration of products like ∫e^x sin(x) dx by organizing repeated integration by parts. It is particularly useful when one function repeats upon differentiation or integration. The method involves creating a table with three columns: signs, derivatives, and integrals.
In the first column, alternating signs are listed, starting with “+”. The second column contains the function to be differentiated repeatedly, here sin(x). The third column contains the function to be integrated repeatedly, which is e^x. We differentiate sin(x) until it repeats or becomes zero. Simultaneously, we integrate e^x the same number of times.
The derivative of sin(x) is cos(x). The derivative of cos(x) is -sin(x), completing one cycle. The integral of e^x is always e^x. The table is constructed as follows:
| Sign | Derivative | Integral |
|---|---|---|
| + | sin(x) | e^x |
| – | cos(x) | e^x |
| + | -sin(x) | e^x |
We multiply diagonally and apply the signs. The integral is obtained by multiplying sin(x) by e^x with a positive sign, cos(x) by e^x with a negative sign. This gives e^x sin(x) – e^x cos(x). Since the original function reappears, we set up an equation to solve for the integral.
The integral ∫e^x sin(x) dx equals e^x sin(x) – e^x cos(x) – ∫e^x sin(x) dx. Adding ∫e^x sin(x) dx to both sides yields 2∫e^x sin(x) dx = e^x sin(x) – e^x cos(x). Dividing by 2 isolates the integral, giving ∫e^x sin(x) dx = (e^x sin(x) – e^x cos(x))/2. Adding the constant of integration provides the final answer: ∫e^x sin(x) dx = (e^x sin(x) – e^x cos(x))/2 + C.
What common mistakes occur while integrating e^x sin(x), and how can they be avoided?
Integrating e^x sin(x) often leads to common errors due to the iterative nature of integration by parts. A frequent mistake is sign errors because incorrect application of the integration by parts formula affects the final result. The integration by parts formula is ∫u dv = uv – ∫v du, and careful tracking of signs is necessary. Incorrect signs lead to an incorrect final answer.
Another common error is failing to complete the second integration by parts. The integral ∫e^x sin(x) dx requires two applications of integration by parts to return to the original integral. Prematurely stopping results in an incomplete and incorrect solution. Completing both iterations is essential.
A third mistake involves incorrectly choosing u and dv. The choice of u affects the complexity of the integral. While either function can be u, consistently choosing the same functions for differentiation and integration is crucial. Switching roles midway through the process introduces confusion and errors.
Forgetting the constant of integration is a common oversight. Every indefinite integral requires the addition of a constant, C, to represent the family of antiderivatives. Omitting C results in an incomplete answer. Always add “+ C” at the end of the integration process. Finally, algebraic errors during simplification can lead to an incorrect final result. Carefully check each step to ensure accuracy.
So, there you have it! Integrating e^x sin(x) might seem daunting at first, but with a little perseverance and those integration techniques, you can totally nail it. Keep practicing, and before you know it, you’ll be integrating like a pro. Happy calculating!