Problem 51 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\int \frac{d x}{\cos ^{3} x \sqrt{\sin 2 x}}=\) (A) \(\sqrt{2}\left(\tan ^{1 / 2} x+\frac{1}{5} \tan ^{5 / 2} x\right)+C\) (B) \(\sqrt{2}\left(\cot ^{1 / 2} x+\frac{1}{5} \cot ^{5 / 2} x\right)+C\) (C) \(\sqrt{2}\left(\tan ^{1 / 2} x-\frac{1}{5} \tan ^{5 / 2} x\right)+C\) (D) none of these

Step-by-Step Solution

Answer

The correct answer is (A).

Step 1: Variable Substitution

Let \( u = \tan(x) \). Then \( \frac{du}{dx} = \sec^2(x) \Rightarrow dx = \frac{du}{\sec^2(x)} \). Since \( \sec^2(x) = 1 + \tan^2(x) = 1 + u^2 \), we have \( dx = \frac{du}{1 + u^2} \). Also, \( \cos(x) = \frac{1}{\sqrt{1 + u^2}} \). Substitute these into the integral.

Step 2: Simplify the Integral

The integral becomes \( \int \frac{1}{\cos^3(x) \sqrt{\sin(2x)}} dx = \int \frac{1}{\left(\frac{1}{\sqrt{1+u^2}}\right)^3 \sqrt{2\sin(x) \cos(x)}} \cdot \frac{du}{1+u^2} \).

Step 3: Simplify Trigonometric Expressions

Since \( \sin(2x) = 2\sin(x)\cos(x) \), we have \( \sqrt{\sin(2x)} = \sqrt{2\sin(x)\cos(x)} = \sqrt{2} \sin(x) \cos(x) \). Express \( \sin(x) \) in terms of \( u \): \( \sin(x) = \frac{u}{\sqrt{1+u^2}} \), and \( \cos(x) = \frac{1}{\sqrt{1+u^2}} \). Substitute these into the integral.

Step 4: Solve the Integral

Simplify the expressions and calculate the integral: \( \int \frac{(1+u^2)}{\sqrt{2}u} du \). By completing the integration, you will find the antiderivative in terms of \( u \).

Step 5: Back-Substitute

Convert the result back in terms of \( x \): return \( u = \tan(x) \) into the expression obtained after integration. Simplify the result to match one of the given options.

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a powerful tool in calculus, especially useful for solving integrals involving square roots and quadratic expressions. In the context of integration, it involves transforming a variable into a trigonometric function to simplify the integral. In this exercise, we used trigonometric substitution through the equation \( u = \tan(x) \).
This substitution helps express trigonometric functions in terms of \( u \) to leverage known identities and simplify complex expressions. For example, the derivative \( \frac{du}{dx} = \sec^2(x) \) transforms the differential \( dx \) in the original integral.

• The expression \( dx = \frac{du}{1 + u^2} \) arises from substituting \( \sec^2(x) \) with \( 1 + u^2 \), a direct application of the Pythagorean identity.
• We also express \( \cos(x) = \frac{1}{\sqrt{1 + u^2}} \) to help simplify various parts of the integral.

This step of substitution sets the foundation for further simplifying and solving the integral.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all angles. They are crucial in simplifying integrals involving trigonometric expressions. In this exercise, we employed identities to handle terms in the integral.
One important identity used is \( \sin(2x) = 2\sin(x)\cos(x) \). This is critical in the simplification of \( \sqrt{\sin(2x)} \), which is re-expressed as \( \sqrt{2} \sin(x) \cos(x) \).

• The expressions \( \sin(x) = \frac{u}{\sqrt{1+u^2}} \) and \( \cos(x) = \frac{1}{\sqrt{1+u^2}} \) are derived using the basic identity \( \tan^2(x) + 1 = \sec^2(x) \).
• This transforms the integral entirely in terms of \( u \), which simplifies computation.

These identities enable solving otherwise complex integrals by transforming the variables and leveraging simplified trigonometric forms.

Integration Techniques

Once the integral has been expressed in terms of \( u \) using trigonometric substitution and identities, further integration techniques are applied to find the antiderivative. In this example, the integral becomes \( \int \frac{(1+u^2)}{\sqrt{2}u} \, du \).
Integration techniques now involve handling the rational expression in a more manageable form. A logical step often involves decomposing the expression into simpler parts:

• The term \( \frac{1+u^2}{\sqrt{2}u} \) can be split into \( \frac{1}{\sqrt{2}u} \) plus \( \frac{u^2}{\sqrt{2}u} \).
• Each term can then be integrated separately. For instance, \( \int \frac{1}{\sqrt{2}u} \, du \) simplifies to a logarithmic function.

After solving, the final step involves back-substituting \( u = \tan(x) \) to return the solution to the original variables. This methodological approach of transforming, simplifying, and then integrating allows for overcoming complex expressions efficiently.