Problem 45 - Thomas Calculus

Question

Evaluate each integral in Exercises \(43-46\) by using trigonometric identities and substitutions to reduce it to standard form. $$ \int \csc x \sin 3 x d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( 5x - \sin 2x + C \).

Step 1: Understand the Integral

The given integral is \( \int \csc x \sin 3x \, dx \). We need to evaluate this by using trigonometric identities and substitutions.

Step 2: Use Trigonometric Identities

Recall that \( \csc x = \frac{1}{\sin x} \). Substitute this identity into the integral: \[ \int \csc x \sin 3x \, dx = \int \frac{1}{\sin x} \cdot \sin 3x \, dx. \] This simplifies to \( \int \frac{\sin 3x}{\sin x} \, dx \).

Step 3: Simplify the Expression

Simplifying \( \frac{\sin 3x}{\sin x} \) using the identity \( \sin 3x = 3 \sin x - 4 \sin^3 x \), we get: \( \frac{3 \sin x - 4 \sin^3 x}{\sin x} = 3 - 4 \sin^2 x \).

Step 4: Substitute and Integrate

So the integral becomes \( \int (3 - 4 \sin^2 x) \, dx \). Split the integrals: \[ \int 3 \, dx - \int 4 \sin^2 x \, dx. \] The first integral is straightforward: \( 3x \).

Step 5: Use Power-Reduction Identity

For the second integral, use the power-reduction identity \( \sin^2 x = \frac{1 - \cos 2x}{2} \). Substitute into the integral: \[ \int 4 \sin^2 x \, dx = \int 4 \left(\frac{1 - \cos 2x}{2}\right) \, dx = \int (2 - 2 \cos 2x) \, dx. \]

Step 6: Integrate Remaining Terms

Integrate the remaining terms: \[ \int 2 \, dx - \int 2 \cos 2x \, dx = 2x - 2 \cdot \frac{1}{2} \sin 2x = 2x - \sin 2x. \] Combine with the previous result: \[ 3x + 2x - \sin 2x = 5x - \sin 2x. \]

Step 7: Combine and Simplify the Results

Combine the results from previous steps to get: \[ \int (3 - 4 \sin^2 x) \, dx = 5x - \sin 2x + C, \] where \( C \) is the constant of integration.

Key Concepts

Trigonometric Substitution

Trigonometric substitution is an essential technique in calculus, especially for dealing with integrals involving trigonometric functions. In this particular exercise, trigonometric substitution plays a crucial role in transforming the integrand into a more standard and solvable form. By substituting known identities, we simplify the original expression: - Replace the cosecant function, so that it involves sine, which is easier to manipulate. - In the given problem, using the identity \[ \csc x = \frac{1}{\sin x} \], allows us to rewrite the integral and simplify it further.The process streamlines solving the integral, all thanks to the clever use of trigonometric identities. This method can simplify a seemingly complex problem into a standard form that is much more straightforward to integrate.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions, which are true for all values of the involved variables. They form the foundation of simplifying and solving integrals as seen in the given exercise. Key identities help in reducing complex trigonometric expressions, making them manageable. Let's take a closer look:- The identity \[ \sin 3x = 3 \sin x - 4 \sin^3 x \] played a pivotal role in this example, expressing a higher multiple angle sine function in terms of simpler powers of sine.- Furthermore, \[ \sin^2 x = \frac{1 - \cos 2x}{2} \] is useful for tackling integrals containing squared sine terms, as it reduces the power and complexity of the expression.By applying these identities at the right moment, the integral takes on a simplified form, easily broken down into parts that are individually solvable.

Calculus Integration Techniques

Integration techniques in calculus are the strategies used to solve integrals, which can often look daunting at first glance. In the exercise provided, a combination of fundamental integration techniques were applied:- Breaking down the integral \[ \int (3 - 4 \sin^2 x) \, dx \] into simpler parts, namely \[ \int 3 \, dx \] and \[ \int 4 \sin^2 x \, dx \].- For straight-forward terms like \[ \int 3 \, dx \], applying basic integration results in \[ 3x \].- For more complex terms such as \[ \int 4 \sin^2 x \, dx \], we apply a power-reduction formula to convert it into a more manageable form.These integration techniques illustrate the diverse approaches available to solve complicated integrals. Each technique, ranging from using simple identities to applying integral decomposition, aligns strategically to simplify and arrive at a solution effectively.