Solving ∫√(t+7) Dt: A Step-by-Step Guide
Are you struggling with integrals? Don't worry, you're not alone! Integrals can seem daunting at first, but with a little practice and the right techniques, you can master them. In this article, we'll break down how to solve the integral ∫√(t+7) dt in a clear and easy-to-understand way. Let's dive in!
Understanding the Integral ∫√(t+7) dt
Before we jump into the solution, let's make sure we understand what we're dealing with. The integral ∫√(t+7) dt is a definite integral, which means we're looking for a function whose derivative is √(t+7). In simpler terms, we need to find the antiderivative of √(t+7).
Integrals are a fundamental concept in calculus and are used in various fields such as physics, engineering, and economics. They help us calculate areas, volumes, and other important quantities. The integral symbol ∫ represents the integration operation, and 'dt' indicates that we're integrating with respect to the variable 't'.
Our main keyword here is the integral ∫√(t+7) dt. This specific integral involves a square root function within the integrand, which adds a slight twist to the problem. However, with a clever substitution, we can simplify it and find the solution. Understanding the components of the integral—the integrand √(t+7) and the differential 'dt'—is crucial for choosing the right integration technique. Recognizing that √(t+7) can be rewritten as (t+7)^(1/2) is the first step towards applying a u-substitution, a common method for solving integrals of this form. The u-substitution method simplifies the integral by replacing a complex expression with a single variable, making it easier to integrate. So, let's move on to the next step and see how we can use this substitution to solve our integral. Stay tuned as we break down each step to ensure you grasp the concept fully.
The U-Substitution Method
The u-substitution method is a powerful technique for solving integrals, especially when the integrand contains a composite function. A composite function is essentially a function within another function, like our √(t+7) where (t+7) is inside the square root. The idea behind u-substitution is to replace this inner function with a new variable, 'u', to simplify the integral. This method often transforms a complex integral into a more manageable form that we can easily solve using basic integration rules.
In our case, the obvious choice for 'u' is the expression inside the square root, which is (t+7). By setting u = t+7, we're essentially simplifying the square root part of the integral. But it's not just about replacing (t+7) with 'u'; we also need to deal with the 'dt' part of the integral. This is where the derivative comes into play. We need to find the derivative of 'u' with respect to 't', which is du/dt. Then, we can solve for 'dt' in terms of 'du', allowing us to substitute both (t+7) and 'dt' in the original integral.
Remember, the goal of u-substitution is to make the integral simpler. By replacing a complex expression with a single variable, we often end up with an integral that we can solve using the power rule or other basic integration techniques. It might seem like a lot of steps, but once you get the hang of it, u-substitution becomes a valuable tool in your integration toolkit. So, let's take the first step and define our 'u' and see how this substitution simplifies our integral ∫√(t+7) dt.
Step-by-Step Solution
Now, let's break down the solution to ∫√(t+7) dt step-by-step. This will make the process clear and easy to follow. We'll be using the u-substitution method, as discussed earlier, to simplify the integral. Here’s how we proceed:
1. Define u
The first step in the u-substitution method is to identify a suitable expression within the integrand to replace with 'u'. In our case, the expression inside the square root, (t+7), seems like a good candidate. So, we let:
u = t + 7
This substitution simplifies the square root part of the integral. Now, we need to find the relationship between 'du' and 'dt'.
2. Find du/dt and Solve for dt
Next, we need to find the derivative of 'u' with respect to 't', which is du/dt. Differentiating both sides of u = t + 7 with respect to 't', we get:
du/dt = 1
Now, we solve for 'dt' in terms of 'du'. Multiplying both sides by 'dt', we have:
du = dt
This is a crucial step because it allows us to replace 'dt' in the original integral with 'du'.
3. Substitute u and du into the Integral
Now that we have defined 'u' and found 'dt' in terms of 'du', we can substitute these into the original integral. The integral ∫√(t+7) dt becomes:
∫√u du
This looks much simpler than the original integral, doesn't it? We've successfully transformed the integral using u-substitution.
4. Rewrite the Square Root as a Power
To make it easier to integrate, we can rewrite the square root of 'u' as a power of 'u'. Recall that √u is the same as u^(1/2). So, our integral becomes:
∫u^(1/2) du
Now, we have a simple power function that we can integrate using the power rule for integration.
5. Apply the Power Rule for Integration
The power rule for integration states that ∫x^n dx = (x^(n+1))/(n+1) + C, where C is the constant of integration. Applying this rule to our integral ∫u^(1/2) du, we get:
(u^((1/2)+1))/((1/2)+1) + C
Simplifying the exponents and fractions:
(u^(3/2))/(3/2) + C
(2/3)u^(3/2) + C
6. Substitute Back for t
We're not done yet! We need to substitute back for 't' since our original integral was in terms of 't'. Recall that we defined u = t + 7. Substituting this back into our result, we get:
(2/3)(t + 7)^(3/2) + C
This is the antiderivative of √(t+7).
7. The Final Answer
So, the solution to the integral ∫√(t+7) dt is:
(2/3)(t + 7)^(3/2) + C
Where C is the constant of integration. This constant is essential because the derivative of a constant is zero, meaning there are infinitely many antiderivatives that differ only by a constant. We've successfully solved the integral using the u-substitution method! Each step was crucial, from defining 'u' to substituting back for 't', to arrive at the final answer. Now, let's summarize the key takeaways from this solution.
Key Takeaways
Let's recap the key steps and concepts we've covered in solving the integral ∫√(t+7) dt. This will help solidify your understanding and make it easier to tackle similar problems in the future.
- U-Substitution: We used the u-substitution method, which is a powerful technique for simplifying integrals involving composite functions. The key is to identify a suitable expression within the integrand to replace with 'u'. In our case, we chose u = t + 7.
- Finding du/dt: After defining 'u', we found the derivative of 'u' with respect to 't' (du/dt) and solved for 'dt' in terms of 'du'. This allowed us to substitute both 'u' and 'du' into the original integral.
- Transforming the Integral: By substituting 'u' and 'du', we transformed the integral ∫√(t+7) dt into ∫√u du, which is much simpler to solve.
- Power Rule for Integration: We rewrote √u as u^(1/2) and applied the power rule for integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C. This gave us (2/3)u^(3/2) + C.
- Substituting Back: Finally, we substituted back for 't' by replacing 'u' with (t + 7), resulting in the final answer: (2/3)(t + 7)^(3/2) + C.
Understanding these key takeaways will not only help you solve integrals like ∫√(t+7) dt but also equip you with the skills to approach a wide range of integration problems. Remember, practice is crucial. The more you solve integrals, the more comfortable you'll become with different techniques and substitutions. So, keep practicing, and you'll master integration in no time!
In summary, solving ∫√(t+7) dt involved a strategic application of the u-substitution method. This method is particularly useful when dealing with composite functions within an integral. By carefully choosing the 'u' (in this case, t+7), we simplified the integral into a manageable form. The subsequent steps, including finding du/dt, substituting back into the original integral, applying the power rule for integration, and finally substituting back for 't', are all crucial components of this technique. Mastering these steps allows for efficient and accurate solutions to a variety of integral problems. Keep practicing, and you'll find that these techniques become second nature, enabling you to tackle more complex integrals with confidence. With a solid grasp of these fundamental concepts, you'll be well-prepared to tackle more advanced calculus topics and applications.
Conclusion
Solving integrals like ∫√(t+7) dt might seem challenging at first, but with the right approach and techniques, it becomes much more manageable. We've walked through the step-by-step solution using the u-substitution method, highlighting the key concepts and steps involved. Remember, practice makes perfect, so keep solving integrals to sharpen your skills.
We hope this guide has been helpful in understanding how to solve this particular integral. If you're looking for more resources on calculus and integration, check out reputable websites like Khan Academy's Calculus section for additional tutorials and practice problems.
