The Value and Importance of Definite Integrals in Mathematics

Definite integrals are a fundamental concept in calculus that allow us to determine the area under a curve, volumes of solids of revolution, and many other valuable applications. In this article, we will explore a specific definite integral and its evaluation through a method of substitution. We'll discuss the process, the mathematics involved, and the significance of these techniques in both theoretical and applied contexts.

The Definite Integral in Question

The given integral is: displaystyle int_{0}^{frac{pi}{2}} frac{cosx}{1 sinx} dx

Evaluation through Method of Substitution

One common method for evaluating definite integrals is the method of substitution, which can simplify complex integrands. Here, we will use this technique to evaluate the given integral step by step.

Let's start by making the substitution:

Let u 1 sinx, which implies du cosx dx.

Now, we rewrite the integral in terms of u:

displaystyle int frac{1}{u} du

By integrating, we get:

displaystyle ln|u| C

Substituting back for u, we get:

displaystyle ln|1 sinx| C

Now, we evaluate the definite integral from 0 to frac{pi}{2}:

displaystyle [ln|1 sinx|]_{0}^{frac{pi}{2}}

Calculating the limits, we get:

displaystyle ln|1 sin(frac{pi}{2})| - ln|1 sin(0)|

Simplifying the expressions:

displaystyle ln(2) - ln(1)

Since ln(1) 0, we get:

displaystyle ln(2)

Thus, the value of the definite integral is:

displaystyle ln(2)

Alternative Method of Substitution

To further illustrate the method, we can also use the alternative substitution:

Let t 1 - sinx, which implies dt -cosx dx.

Now, we rewrite the integral in terms of t:

displaystyle int frac{-1}{t} dt

By integrating, we get:

displaystyle -ln|t| C

Substituting back for t, we get:

displaystyle -ln|1 - sinx| C

Now, we evaluate the definite integral from 0 to frac{pi}{2}:

displaystyle [ -ln|1 - sinx| ]_{0}^{frac{pi}{2}}

Calculating the limits, we get:

displaystyle -ln|1 - sin(frac{pi}{2})| ln|1 - sin(0)|

Simplifying the expressions:

displaystyle -ln(0) ln(1)

Since ln(0) is undefined, this method must be adjusted considering the behavior of the function near x frac{pi}{2}.

However, the first method of substitution remains valid and simpler for this particular integral.

Significance of Method of Substitution

The process of method of substitution is not only a mathematical technique but also a powerful tool for solving a wide range of problems in mathematics and science. It is used in various fields such as physics, engineering, and economics, where definite integrals play a pivotal role.

In physics, definite integrals help in determining quantities like the work done by a force. In engineering, they are used to calculate areas, volumes, and lengths of curves. In economics, they can be used to model and optimize various systems.

Understanding and mastering the method of substitution is essential for anyone pursuing advanced studies in mathematics, science, or related fields.

Conclusion

In conclusion, the value of the definite integral displaystyle int_{0}^{frac{pi}{2}} frac{cosx}{1 sinx} dx is ln(2). This result is obtained through the method of substitution, where appropriate substitutions significantly simplify the integrand. The method of substitution is a versatile and powerful technique in calculus, and its importance extends beyond this specific integral to a wide range of mathematical and real-world applications.