Integrate E^(2x+5): Step-by-Step Solution & Check

by ADMIN 50 views

Hey guys! Today, we're diving into the exciting world of integral calculus, specifically focusing on integrating exponential functions. We'll take a close look at the integral ∫e^(2x+5) dx, breaking down each step and, most importantly, verifying our solution through differentiation. This ensures we not only get the answer but also understand why it's the right answer. So, let's get started!

When tackling exponential integrals, the key is recognizing the basic form and applying the appropriate techniques. In this case, we have ∫e^(2x+5) dx. The exponent 2x+5 is a linear expression, which means we can use a simple u-substitution to simplify the integral. Let's dive deeper into this method.

U-Substitution: The Golden Rule

The method of u-substitution is a powerful tool for simplifying integrals. It's essentially the reverse of the chain rule in differentiation. Here's how it works:

  1. Identify a suitable 'u': Look for a function within the integral whose derivative is also present (or a constant multiple of it). In our case, the exponent 2x+5 looks like a good candidate. Let u = 2x + 5.
  2. Calculate du: Find the derivative of u with respect to x. du/dx = 2. This means du = 2 dx.
  3. Adjust the integral: We need to express the original integral entirely in terms of u and du. Notice that we have dx in the original integral, but we have 2 dx in our expression for du. So, we can rewrite dx as (1/2) du.
  4. Substitute and integrate: Replace 2x+5 with u and dx with (1/2) du in the integral. This gives us ∫e^u (1/2) du.
  5. Solve the simpler integral: Now we have a much simpler integral! (1/2) ∫e^u du. The integral of e^u with respect to u is simply e^u. So, we have (1/2)e^u + C, where C is the constant of integration.
  6. Substitute back: Finally, replace u with its original expression, 2x+5. This gives us our final result: (1/2)e^(2x+5) + C.

So, after applying the u-substitution method, we've found that the integral of e^(2x+5) dx is (1/2)e^(2x+5) + C. But how do we know this is correct? This is where differentiation comes in handy. Verifying our solution is a crucial step in calculus, ensuring accuracy and a solid understanding of the concepts involved. So, let's move on to the next section and check our work.

One of the beautiful things about calculus is the inverse relationship between integration and differentiation. Differentiation allows us to verify the accuracy of our integration. If we differentiate the result we obtained from integration, we should get back the original function. So, let's put this to the test!

Our proposed solution for the integral ∫e^(2x+5) dx is (1/2)e^(2x+5) + C. To verify this, we'll differentiate (1/2)e^(2x+5) + C with respect to x. Remember, the derivative of a constant is zero, so the + C term will disappear upon differentiation. The core of our differentiation will involve the chain rule. The chain rule is essential when differentiating composite functions. It states that the derivative of f(g(x)) is f'(g(x)) * g'(x). In simpler terms, we differentiate the outer function while keeping the inner function the same, and then multiply by the derivative of the inner function.

Applying the chain rule here, we have:

d/dx [(1/2)e^(2x+5) + C] = (1/2) * d/dx [e^(2x+5)] + d/dx [C]

Since the derivative of a constant C is 0, the equation is simplified as:

= (1/2) * d/dx [e^(2x+5)]

Here, our outer function is e^u and our inner function is u = 2x + 5. The derivative of e^u with respect to u is simply e^u. The derivative of 2x+5 with respect to x is 2. Applying the chain rule, we get:

(1/2) * e^(2x+5) * d/dx(2x+5)

= (1/2) * e^(2x+5) * 2

The 2 in the numerator and the (1/2) cancel out, leaving us with:

= e^(2x+5)

And there you have it! We differentiated (1/2)e^(2x+5) + C and got back e^(2x+5), which is the original integrand. This confirms that our integration was performed correctly. Guys, this process of verifying through differentiation is super important. It's not just about getting the right answer; it's about building confidence in your calculus skills. You know your answer is rock solid when the derivative of your integral brings you back to where you started.

While u-substitution is the most straightforward approach for this integral, let's briefly touch upon an alternative perspective and some common mistakes to avoid. An alternative approach might involve recognizing the form of the integral and directly applying the formula for the integral of e^(ax+b), which is (1/a)e^(ax+b) + C. In our case, a = 2 and b = 5, so this formula directly gives us (1/2)e^(2x+5) + C. However, even though this method is quicker, it's essential to understand the underlying principles of u-substitution, as it's a versatile technique applicable to a wider range of integrals.

One common pitfall when integrating exponential functions is forgetting the constant of integration, C. Remember, the indefinite integral represents a family of functions, all differing by a constant. Omitting C means you're only giving one specific antiderivative, not the general solution. Another common mistake is incorrectly applying the chain rule (or rather, its reverse) during integration. For instance, some might forget to divide by the derivative of the inner function (in our case, dividing by 2). This is where writing out the u-substitution steps explicitly can help prevent errors. It's also vital to double-check your work by differentiating, as we've demonstrated. If your derivative doesn't match the original integrand, something went wrong, and you need to revisit your steps. Practice is key to mastering these techniques and avoiding common pitfalls. The more integrals you solve, the more comfortable you'll become with identifying the appropriate methods and executing them accurately.

Okay, guys, let's recap the key takeaways from our exploration of ∫e^(2x+5) dx. We successfully integrated the exponential function using u-substitution, a fundamental technique in integral calculus. We then rigorously verified our result through differentiation, highlighting the crucial inverse relationship between these two operations. We also discussed an alternative approach using a direct formula and pointed out common pitfalls to avoid, such as forgetting the constant of integration or misapplying the chain rule. But our journey doesn't end here! Calculus is a vast and fascinating field, and there's always more to learn.

If you want to deepen your understanding of integration, I encourage you to explore more complex integrals involving exponential functions, trigonometric functions, and other types of functions. Try tackling integrals that require multiple substitutions or integration by parts. Practice is absolutely crucial for mastering these techniques. You might also want to investigate definite integrals, which involve evaluating the integral over a specific interval and provide a numerical result. Definite integrals have numerous applications in physics, engineering, and other fields. Furthermore, consider exploring the connection between integration and differential equations. Many real-world phenomena can be modeled using differential equations, and integration plays a vital role in solving them. So, keep practicing, keep exploring, and never stop questioning. The world of calculus is full of exciting challenges and rewarding discoveries!

In conclusion, evaluating the integral ∫e^(2x+5) dx provides a great example of how to approach integrating exponential functions. We saw the power of u-substitution in simplifying complex integrals and the importance of differentiation in verifying our results. Guys, remember to always check your work, be mindful of common pitfalls, and never stop exploring the fascinating world of calculus. Until next time, happy integrating!