Question

Integrals with ${\sin }^{2}x$ and ${\cos }^{2}x$ Evaluate the following integrals. $$\int {\sin }^2(\theta +\frac{\pi }{6})d\theta$$

Short answer

Question: Evaluate the integral $\int {\sin }^2(\theta +\frac{\pi }{6})d\theta$. Answer: $\frac{1}{2}(\theta -\frac{1}{2}\sin (2\theta +\frac{\pi }{3}))+C$

Step-by-step solution

Rewrite using trigonometric identity

Rewrite the sine squared function using the identity ${\sin }^{2}x=(1-\cos 2x)/2$. This results in: $$\int {\sin }^2(\theta +\frac{\pi }{6})d\theta =\int \frac{1-\cos [2(\theta +\frac{\pi }{6})]}{2}d\theta$$

Distribute the argument

Distribute the argument inside the cosine function: $$\int \frac{1-\cos (2\theta +\frac{\pi }{3})}{2}d\theta$$

Integrate the function

Integrate each term separately. The integral of constant 1 is $\theta$ and the integral of $\cos (k\theta )$ is $(1/k)\sin (k\theta )$. Therefore, our integral becomes: $$\frac{1}{2}\int (1-\cos (2\theta +\frac{\pi }{3}))d\theta =\frac{1}{2}(\theta -\frac{1}{2}\sin (2\theta +\frac{\pi }{3}))+C$$ where C is the integration constant.

Final answer

The final result is: $$\frac{1}{2}(\theta -\frac{1}{2}\sin (2\theta +\frac{\pi }{3}))+C$$

Key concepts

Trigonometric Identities

Understanding trigonometric identities is crucial when working with integrals involving trigonometric functions. An identity is a tautology, a mathematical equation that is always true for any value that meets the equation's conditions. In the context of integrating the sine squared function, one commonly used identity is \[\begin{equation}\text{sin}^2(x) = \frac{1 - \text{cos}(2x)}{2}\text{,}\text{\}\end{equation}\] which helps to simplify the integral by reducing the power of the sine function, making the integration process manageable. This specific identity is derived from the more general double-angle formula for cosine, \[\begin{equation}\text{cos}(2x) = \text{cos}^2(x) - \text{sin}^2(x)\text{,}\text{\}\end{equation}\] which, through algebraic manipulation using the Pythagorean identity \[\begin{equation}\text{sin}^2(x) + \text{cos}^2(x) = 1\text{,}\text{\}\end{equation}\] gives us the desired form for integration. Familiarity with these identities allows for effective translation of trigonometric integral expressions into simpler forms that can be more easily integrated.

Integral Calculus

Integral calculus is a branch of mathematical analysis that deals with the sizes and properties of geometric figures and irregular shapes. It is mainly focused on the concept of an integral, which can be thought of as an accumulation function or a generalized sum. There are many types of integrals, but in our context, we are handling definite and indefinite integrals of functions involving trigonometric expressions. In essence, when we integrate a function, we are finding the area under the curve of the function's graph within a certain interval, or in the case of an indefinite integral, we are finding the anti-derivative or the original function whose derivative gives us the integrand. Learning integration techniques, such as substitution, integration by parts, and identifying useful trigonometric identities, is vital for solving integrals effectively. This foundational knowledge allows us to tackle a wide variety of problems within mathematics and applied fields.

Sine Squared Function

The sine squared function, denoted as \[\begin{equation}\text{sin}^2(x)\text{,}\text{\}\end{equation}\]is a power function of the basic trigonometric function sine. It represents the square of the sine of an angle \[\begin{equation}x\text{.}\text{\}\end{equation}\]This function oscillates between 0 and 1 and has a period of \[\begin{equation}\text{π}\text{.}\text{\}\end{equation}\]Sine functions, and by extension, sine squared functions, are frequently encountered in various fields such as physics, engineering, and signal processing. In integration, it is often necessary to convert a sine squared function into a more manageable form to simplify the integration process. This is commonly done using trigonometric identities, as it reduces the complexity of the integral and allows for the use of more straightforward integration methods.

Integration Techniques

Integration techniques are various methods used to calculate the integral or anti-derivative of a function. In the realm of trigonometric functions, some commonly used techniques include:

• Substitution: Changing the variable of integration to simplify the integral.
• Integration by parts: Based on the product rule of differentiation, used for functions that are products of two simpler functions.
• Trigonometric substitution: Involves using trigonometric identities to reduce complex expressions to simpler forms.
• Partial fractions: Decomposing rational functions into simpler fractions that are easier to integrate.

In our example, we specifically used a trigonometric identity to transform the sine squared function. Breaking down complex trigonometric functions into more manageable parts allows for easier integration and leads to more immediate results. The ability to recognize which technique to use comes with practice and familiarity with different types of functions and their respective integration strategies.