Problem 62 - Calculus Single Variable

Question

Calculate the integrals. $$ \int_{0}^{\pi / 3} \frac{\sin (x)}{1-\sin ^{2}(x)} d x $$

Step-by-Step Solution

Answer

The value of the integral is 1.

Step 1: Simplify the Denominator

Observe that the denominator can be rewritten using the identity \(1 - \sin^2(x) = \cos^2(x)\). So, the integral becomes: \[ \int_{0}^{\pi/3} \frac{\sin(x)}{\cos^2(x)} \, dx \] This can be further simplified to: \[ \int_{0}^{\pi/3} \tan(x) \sec(x) \, dx \]

Step 2: Identify a Suitable Substitution

Notice that the integral \(\int \tan(x) \sec(x) \, dx\) is a standard form. We can use substitution by letting \ u = \sec(x) \, making \ du = \sec(x)\tan(x) \, dx \, which matches the integrand.

Step 3: Change the Limits of Integration

Convert the limits from \(x\) to \(u\). When \ x = 0 \, \ \sec(0) = 1 \. When \ x = \frac{\pi}{3} \, \ \sec\left(\frac{\pi}{3}\right) = 2 \. Hence, the limits change from \ x: [0, \frac{\pi}{3}] \ to \ u: [1, 2] \.

Step 4: Evaluate the Integral

The integral now is: \[ \int_{1}^{2} \, du \] The evaluation of this integral is straightforward: \[ u = [u]_{1}^{2} = 2 - 1 = 1 \\]

Step 5: Conclude the Solution

Thus, the integral evaluates to 1 after applying the substitution, changing limits, and performing the final integration.

Key Concepts

Definite Integrals

Definite integrals are a core concept in Integral Calculus that help determine the area under a curve within a specific interval. In essence, they take the integral of a function and evaluate it from one point, called the lower limit, to another, known as the upper limit.
This evaluation results in a specific numerical value rather than a function, indicating the net area between the graph of the function and the x-axis.

To calculate a definite integral, follow these general steps:

• Identify the function you need to integrate.
• Integrate the function to find the antiderivative.
• Evaluate the antiderivative at the upper limit of integration and subtract the evaluation at the lower limit.

In our exercise, the function of interest \[ \int_{0}^{\pi / 3} \frac{\sin(x)}{1 - \sin^2(x)} \, dx \] is integrated over an interval from 0 to \( \frac{\pi}{3} \). By evaluating the antiderivative at the limits, we find the area under the curve between these points.

Trigonometric Substitution

Trigonometric substitution is a powerful technique used to simplify integrals involving square roots and trigonometric identities.
This method involves replacing variables with trigonometric functions to transform the integral into a more manageable form.

In our example, the integral \[ \int_{0}^{\pi/3} \frac{\sin(x)}{1 - \sin^2(x)} \, dx \] was simplified by recognizing the identity \( 1 - \sin^2(x) = \cos^2(x) \).
This allowed us to rewrite the integral as \[ \int_{0}^{\pi/3} \tan(x) \sec(x) \, dx \], making it easier to handle.

The substitution used here leveraged the relationship between trigonometric functions to simplify evaluation. It involves these steps:

• Identify a trigonometric identity that matches the form of the integral.
• Make a substitution to simplify the integral, often setting a new variable equal to the trigonometric function.
• Change the limits if it's a definite integral and solve.

Hence, trigonometric substitution is vital in solving and transforming complicated integrals involving trigonometric functions.

Integration Techniques

Integral Calculus offers a variety of techniques to solve integrals effectively. Choosing the right technique is key to simplifying an integral and computing its value efficiently.

Some common integration techniques include:

• Substitution: Simplifying the integral by introducing a new variable.
• Integration by Parts: A method based on the product rule of differentiation, useful when dealing with products of functions.
• Trigonometric Substitution: Used when dealing with integrals involving square roots or quadratics, as seen in our example.
• Partial Fraction Decomposition: Breaking down complex fractions into simpler parts that can be integrated easily.

In our specific case, a suitable substitution was made by setting \( u = \sec(x) \), which corresponded perfectly to the integrand \( \tan(x) \sec(x) \, dx \).
By using substitution, the integral \[ \int_{0}^{\pi/3} \tan(x) \sec(x) \, dx \] was transformed into the simpler form of \[ \int_{1}^{2} \, du \], making evaluation straightforward.

Among these techniques, substitution is often the simplest and most efficient way to approach an integral, especially when it reduces a complex expression into a basic one, as seen in the step-by-step solution.