Solving A Definite Integral: A Step-by-Step Guide

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Hey math enthusiasts! Today, we're diving into the world of calculus and tackling a definite integral. Specifically, we're going to evaluate: ∫0Ο€23sin⁑2(x+Ο€4) dx{ \int_{0}^{\frac{\pi}{2}} \frac{3}{\sin^2\left(x+\frac{\pi}{4}\right)} \, dx }

This might look a bit intimidating at first glance, but trust me, we'll break it down step by step and make it understandable. This guide is crafted to help you grasp the core concepts of integration, and the problem-solving strategies, ensuring you not only find the solution, but also deepen your understanding of the underlying principles. Let's get started!

Understanding the Problem: Definite Integrals

First off, let's make sure we're all on the same page about what a definite integral is. In essence, a definite integral represents the area under a curve between two specific points (the limits of integration) on the x-axis. In our case, the curve is defined by the function 3sin⁑2(x+Ο€4){ \frac{3}{\sin^2\left(x+\frac{\pi}{4}\right)} }, and we're interested in the area from x=0{x = 0} to x=Ο€2{x = \frac{\pi}{2}}. These limits define the exact interval over which we are calculating the area. The presence of the sin⁑2{\sin^2} term might make you think of trigonometric identities, and you'd be right on the money. Also, keep in mind the constant multiplier (3) in the numerator, as it will simply multiply our final result. The correct application of the power rule and the chain rule are essential to properly evaluate the integral, along with correct substitution and trigonometric formulas.

We are looking to find the accumulated value of the function over the given interval. This is in contrast to indefinite integrals, which give a family of functions (plus a constant). Furthermore, the definite integral gives a single numerical answer (a number) representing the area under the curve between the specified limits. Therefore, the approach involves finding the antiderivative of the function (also called the indefinite integral), and evaluating it at the upper and lower limits of the integration. This gives the values of the antiderivative at the end points of the integration interval. Finally, we subtract the value of the antiderivative at the lower limit from its value at the upper limit to get the final result. In short, remember that evaluating a definite integral involves finding the antiderivative, applying the limits of integration, and subtracting the results. We will break this process into manageable steps to easily solve this integral. Remember to apply all the rules correctly, with proper attention to detail, so you will arrive at the correct answer. The more integrals you solve, the more practice you will get and the faster and easier it will become.

Importance of Trigonometric Identities in Integration

Before we begin, a quick refresher on trigonometric identities can make our lives much easier. Specifically, we'll likely need to use the following:

  • Reciprocal Identities: Remembering that csc⁑(x)=1sin⁑(x){\csc(x) = \frac{1}{\sin(x)}} is crucial. Our integral contains sin⁑2{\sin^2}, implying the potential of using the co-function. The reciprocal identity is essential for rewriting the original integral into a more manageable form. Specifically, 1sin⁑2(x){\frac{1}{\sin^2(x)}} equals csc⁑2(x){\csc^2(x)}.
  • Chain Rule in Reverse: We will need to keep this in mind when integrating. We might see an expression like sin⁑(ax+b){\sin(ax + b)}, and we'll need to remember that the derivative of the inside function (i.e., ax+b{ax + b}) will impact the integral. We need to remember that the chain rule is used when finding the antiderivative of composite functions, and it is a key component to accurately solve the integral.

These identities will help simplify our integral and make it easier to solve. Also, it is crucial to properly memorize the derivative and integral of basic trigonometric functions, like sine, cosine, tangent and their reciprocal counterparts. The knowledge and ability to use them is essential for correctly solving integration problems. Remember that with practice, you'll become more comfortable recognizing these patterns and applying the appropriate identities.

Step-by-Step Solution of the Definite Integral

Let's roll up our sleeves and solve the integral step-by-step. The key here is to break down the problem into smaller, more manageable parts. We'll show you the exact methodology, so you can apply this to other similar problems in the future. Here's our game plan:

  1. Rewrite the Integral: Start by rewriting the integral using the reciprocal identity. This is going to make the integration process easier, because it directly links the integral to the derivative of cotangent. Also, we can take the constant out of the integral, so we can focus on integrating the remaining terms.
  2. Apply Substitution: Use u-substitution to simplify the integral. This often makes the integration process straightforward. Here, we can choose a proper substitution for the angle inside the sine function. Remember to find the derivative of your substitution and use it properly.
  3. Integrate: Find the antiderivative of the simplified integral. Make sure you are paying attention to the formulas of trigonometric functions, as it is very easy to make mistakes in this step.
  4. Back-Substitute: Replace u{u} with its original expression in terms of x{x}.
  5. Apply the Limits of Integration: Evaluate the antiderivative at the upper and lower limits and subtract. This gives us the final value of the definite integral.

Let's dive into each step with detail.

1. Rewriting the Integral

First, let's rewrite the integral using the reciprocal identity for sine: 1sin⁑2(x)=csc⁑2(x){\frac{1}{\sin^2(x)} = \csc^2(x)}. Our integral becomes:

∫0Ο€23csc⁑2(x+Ο€4) dx{ \int_{0}^{\frac{\pi}{2}} 3 \csc^2\left(x+\frac{\pi}{4}\right) \, dx }

At this step, we can bring the constant out of the integral:

3∫0Ο€2csc⁑2(x+Ο€4) dx{ 3 \int_{0}^{\frac{\pi}{2}} \csc^2\left(x+\frac{\pi}{4}\right) \, dx }

2. Applying Substitution

Now, let's use u-substitution to simplify things. Let:

  • u=x+Ο€4{u = x + \frac{\pi}{4}}
  • Then, du=dx{du = dx}

Our limits of integration also need to be adjusted accordingly:

  • When x=0{x = 0}, u=0+Ο€4=Ο€4{u = 0 + \frac{\pi}{4} = \frac{\pi}{4}}
  • When x=Ο€2{x = \frac{\pi}{2}}, u=Ο€2+Ο€4=3Ο€4{u = \frac{\pi}{2} + \frac{\pi}{4} = \frac{3\pi}{4}}

Our integral transforms to:

3βˆ«Ο€43Ο€4csc⁑2(u) du{ 3 \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \csc^2(u) \, du }

3. Integrating

Now, let's integrate csc⁑2(u){\csc^2(u)}. Remember that the integral of csc⁑2(u){\csc^2(u)} is βˆ’cot⁑(u){-\cot(u)}. Thus, we have:

3[βˆ’cot⁑(u)]Ο€43Ο€4{ 3 [-\cot(u)]_{\frac{\pi}{4}}^{\frac{3\pi}{4}} }

4. Back-Substituting (Not Needed in this case)

Since we are using the variable u{u} to perform our integration, we do not need to substitute the original value of x{x} for u{u} because we will be using the limits of integration in terms of u{u}.

5. Applying the Limits of Integration

Finally, we evaluate the antiderivative at the limits of integration and subtract:

3[βˆ’cot⁑(3Ο€4)βˆ’(βˆ’cot⁑(Ο€4))]{ 3\left[-\cot\left(\frac{3\pi}{4}\right) - \left(-\cot\left(\frac{\pi}{4}\right)\right)\right] }

Recall that:

  • cot⁑(3Ο€4)=βˆ’1{\cot\left(\frac{3\pi}{4}\right) = -1}
  • cot⁑(Ο€4)=1{\cot\left(\frac{\pi}{4}\right) = 1}

Therefore:

3[βˆ’(βˆ’1)βˆ’(βˆ’1)]{ 3[ -(-1) - (-1)] }

3[1+1]=3(2)=6{ 3[1 + 1] = 3(2) = 6 }

So, the final answer is 6.

Conclusion: The Final Answer

And there you have it, guys! We have successfully evaluated the definite integral:

∫0Ο€23sin⁑2(x+Ο€4) dx=6{ \int_{0}^{\frac{\pi}{2}} \frac{3}{\sin^2\left(x+\frac{\pi}{4}\right)} \, dx = 6 }

We started with a seemingly complex integral, but by using trigonometric identities, u-substitution, and a step-by-step approach, we were able to find the solution. The most important thing is to understand the concepts and not to get scared by the problem. Remember, practice is key. Keep working through problems, and you'll become more confident in your calculus abilities. Keep going, and happy integrating!

Tips for Success in Integration

To become a master of integration, consider these tips:

  • Practice Regularly: The more you practice, the better you'll become. Work through a variety of problems to get comfortable with different techniques. The more time you spend practicing the different methodologies to solve these problems, the better you will get, and the faster you will become at correctly answering them.
  • Master Basic Integrals: Memorize the basic integral formulas. This will save you time and help you solve more complex problems efficiently. Knowing these will form the foundations of your success in calculus.
  • Understand Trigonometric Identities: Familiarize yourself with trigonometric identities. They are often essential for simplifying integrals. Practice using these trigonometric formulas so that you can correctly apply them.
  • Use U-Substitution Effectively: U-substitution is your friend. Learn how to identify when and how to use it to simplify integrals. Correct application of the technique is key.
  • Check Your Work: Always double-check your work, especially when applying the limits of integration. This will help you catch any mistakes you might have made in your calculations. Checking your work will ensure that you have found the correct answer.
  • Don't Give Up: Integration can be challenging, but don't get discouraged. Keep practicing, and you'll eventually master it. Keep working through problems and with each correct answer, you will grow in confidence.

By following these tips and practicing regularly, you'll be well on your way to mastering definite integrals and conquering the world of calculus! Keep up the great work, and don't hesitate to revisit these steps if you get stuck. Calculus can be challenging, but it can also be very rewarding. Happy integrating, and enjoy the journey!