In calculus, determining the integral of e^(2x) is a fundamental task. Two approaches—differentiation and substitution—offer distinct yet effective ways to find this integral. Let’s explore both methods succinctly.
Method 1: Integral by Differentiation
Using the fundamental theorem of calculus, we start by finding the derivative of e^(2x).
Applying the chain rule yields e^(2x) = 2e^(2x).
Dividing by 2 and integrating, we get ∫ e^(2x) dx = (e^(2x))/2 + C.
Method 2: Integral by Substitution
Through substitution (letting 2x = u), the integral becomes ∫ e^(u) (du/2) = (1/2) ∫ e^(u) du.
Using the integral of e^x, this simplifies to (1/2) e^(2x) + C.
Both methods demonstrate the versatility of calculus in solving problems. Whether through differentiation or substitution, mathematicians can efficiently find the integral of e^(2x), showcasing the interconnectedness of calculus principles.
Understanding how to find the integral of exponential functions is a fundamental skill in calculus. In this step-by-step guide, we will explore the process of finding the integral of the function e^(2x). By following these steps, you’ll be able to master the technique and apply it to similar problems.
Step 1: Identify the Function and Variable
For the given function, e^(2x),
e is the base of the natural logarithm, and x is the variable.
Step 2: Recall the Integral Rules
To find the integral of e^(2x), we’ll use the basic integral rules.
The integral of e^(kx) with respect to x is (1/k) * e^(kx), where k is a constant.
In this case, k is 2.
Recommended Reading: Integral of sin^2x: A Step-by-Step Guide
Step 3: Apply the Integral Rule
Using the integral rule, we can write the integral of e^(2x) as (1/2) * e^(2x).
This is the antiderivative of e^(2x) with respect to x.
Step 4: Add the Constant of Integration
When finding an indefinite integral, always add a constant of integration, denoted by ‘C.’
Therefore, the final result is (1/2) * e^(2x) + C.
Step 5: Check Your Answer
To confirm the correctness of your result, you can differentiate the obtained antiderivative.
The derivative of (1/2) * e^(2x) + C with respect to x should be equal to e^(2x).
Conclusion
With this you’ve found the integral of e^(2x). This process involves recognizing the function, applying integral rules, and adding the constant of integration. Mastering these steps will not only help you solve similar problems but also enhance your understanding of calculus concepts. Keep practicing to improve your skills and build confidence in tackling more complex integrals.
We hope this article has been helpful. If you have any questions, please feel free to comment below.
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