Start with the Weierstrass substitution

Oct 13, 2025 by miodeya.com 147 views

Introduction

Calculus, a branch of mathematics that deals with the study of continuous change, is a fundamental subject in mathematics. It has numerous applications in various fields, including physics, engineering, and economics. One of the most important concepts in calculus is integration, which is used to find the area under curves and the accumulation of quantities. However, there are cases where integration can be done without actually integrating, i.e., without writing the function in terms of a single trigonometric function. In this article, we will explore how to integrate the given function without actually integrating it.

The Problem

The given function is:

\int_0^{\pi/2} \frac{dx}{(\sin x+ \cos x)^2}

This function appears to be a simple trigonometric integral, but it is not as straightforward as it seems. The function involves both sine and cosine functions, which makes it challenging to integrate directly. However, we can use a clever technique to integrate this function without actually integrating it.

Weierstrass Substitution

The Weierstrass substitution is a technique used to integrate functions that involve trigonometric functions. It involves substituting the trigonometric function with a new variable, which makes the integration process easier. In this case, we can use the Weierstrass substitution to integrate the given function.

Let's start by substituting $\tan \frac{x}{2} = t$ . This substitution is useful because it allows us to express the trigonometric functions in terms of the new variable $t$ . We can then use the chain rule to find the derivative of the new variable $t$ with respect to $x$ .

\frac{dt}{dx} = \frac{1}{2} \sec^2 \frac{x}{2}

Now, we can substitute the new variable $t$ into the original function:

\int_0^{\pi/2} \frac{dx}{(\sin x+ \cos x)^2} = \int_0^1 \frac{2}{(1+t^2)^2} dt

This substitution has simplified the integration process, but we still need to integrate the function with respect to $t$ . However, we can use another technique to integrate this function without actually integrating it.

Trigonometric Identity

The given function can be expressed in terms of a trigonometric identity. We can use the identity $\sin^2 x + \cos^2 x = 1$ to rewrite the function:

\int_0^{\pi/2} \frac{dx}{(\sin x+ \cos x)^2} = \int_0^{\pi/2} \frac{dx}{\sin^2 x + 2 \sin x \cos x + \cos^2 x}

Now, we can use the identity $\sin^2 x + \cos^2 x = 1$ to simplify the function:

\int_0^{\pi/2} \frac{dx}{(\sin x+ \cos x)^2} = \int_0^{\pi/2} \frac{dx}{1 + 2 \sin x \cos x}

This function can be integrated using the substitution $u = \sin x \cos x$ . This substitution is useful because it allows us to express the function in terms of a single trigonometric function.

Substitution

Let's start by substituting $u = \sin x \cos x$ . This substitution is useful because it allows us to express the function in terms of a single trigonometric function.

\frac{du}{dx} = \cos^2 x + \sin^2 x = 1

Now, we can substitute the new variable $u$ into the original function:

\int_0^{\pi/2} \frac{dx}{1 + 2 \sin x \cos x} = \int_0^1 \frac{du}{1 + 2u}

This substitution has simplified the integration process, but we still need to integrate the function with respect to $u$ . However, we can use another technique to integrate this function without actually integrating it.

Partial Fractions

The given function can be expressed in terms of partial fractions. We can use the partial fraction decomposition to rewrite the function:

\int_0^{\pi/2} \frac{dx}{1 + 2 \sin x \cos x} = \int_0^1 \frac{du}{1 + 2u} = \int_0^1 \left( \frac{1}{1 + 2u} - \frac{1}{1 + 2u + 2} \right) du

Now, we can integrate the function using the partial fraction decomposition:

\int_0^{\pi/2} \frac{dx}{1 + 2 \sin x \cos x} = \left[ \frac{1}{2} \ln (1 + 2u) - \frac{1}{2} \ln (1 + 2u + 2) \right]_0^1

This function can be evaluated using the fundamental theorem of calculus.

Conclusion

In this article, we have explored how to integrate the given function without actually integrating it. We have used various techniques, including the Weierstrass substitution, trigonometric identities, substitution, and partial fractions, to simplify the integration process. The final answer is:

\int_0^{\pi/2} \frac{dx}{(\sin x+ \cos x)^2} = \frac{1}{2} \ln 3

Introduction

In our previous article, we explored how to integrate the given function without actually integrating it. We used various techniques, including the Weierstrass substitution, trigonometric identities, substitution, and partial fractions, to simplify the integration process. In this article, we will answer some of the most frequently asked questions about integrating without integrating.

Q: What is integrating without integrating?

A: Integrating without integrating refers to the process of finding the antiderivative of a function without actually writing the function in terms of a single trigonometric function. This can be done using various techniques, including the Weierstrass substitution, trigonometric identities, substitution, and partial fractions.

Q: Why is integrating without integrating useful?

A: Integrating without integrating is useful because it allows us to find the antiderivative of a function without actually writing the function in terms of a single trigonometric function. This can be particularly useful when dealing with complex functions that involve multiple trigonometric functions.

Q: What are some common techniques used for integrating without integrating?

A: Some common techniques used for integrating without integrating include:

Weierstrass substitution: This technique involves substituting a trigonometric function with a new variable, which makes the integration process easier.
Trigonometric identities: These are mathematical formulas that relate trigonometric functions to each other. They can be used to simplify the integration process.
Substitution: This technique involves substituting a function with a new variable, which makes the integration process easier.
Partial fractions: This technique involves expressing a function as a sum of simpler functions, which makes the integration process easier.

Q: How do I choose the right technique for integrating without integrating?

A: Choosing the right technique for integrating without integrating depends on the specific function you are trying to integrate. Here are some general tips:

Start with the Weierstrass substitution: This technique is often the most straightforward and easiest to use.
Use trigonometric identities: These can be used to simplify the integration process and make it easier to find the antiderivative.
Try substitution: This technique can be used to simplify the integration process and make it easier to find the antiderivative.
Use partial fractions: This technique can be used to express the function as a sum of simpler functions, which makes the integration process easier.

Q: What are some common mistakes to avoid when integrating without integrating?

A: Some common mistakes to avoid when integrating without integrating include:

Not using the correct technique: Make sure to choose the right technique for the specific function you are trying to integrate.
Not simplifying the function enough: Make sure to simplify the function as much as possible before trying to integrate it.
Not checking your work: Make sure to check your work carefully to ensure that you have found the correct antiderivative.

Q: How do I practice integrating without integrating?

A: Practicing integrating without integrating is essential to becoming proficient in this technique. Here are some tips:

Start with simple functions: Begin with simple functions and gradually move on to more complex ones.
Use online resources: There are many online resources available that can help you practice integrating without integrating.
Work with a partner: Working with a partner can be a great way to practice integrating without integrating and get feedback on your work.
Take your time: Don't rush through the integration process. Take your time and make sure to simplify the function as much as possible before trying to integrate it.

Conclusion

Integrating without integrating is a powerful technique that can be used to find the antiderivative of a function without actually writing the function in terms of a single trigonometric function. By using the right technique and simplifying the function as much as possible, you can make the integration process easier and more efficient. Remember to practice regularly and take your time to ensure that you have found the correct antiderivative.