91Ó°ÊÓ

Trigonometric integration is a method used to evaluate integrals that involve trigonometric functions such as sine, cosine, and tangent. In these integrals, trigonometric identities can often help simplify the expression before integration.
A common goal in trigonometric integration is to convert products of trigonometric functions into sums or differences, making them easier to integrate.
Different formulas, like the product-to-sum identities, play a vital role in this simplification process. By understanding these identities, you can tackle a wide range of trigonometric integrals with more ease and confidence.

The product-to-sum identities are a set of trigonometric formulas that help simplify the product of two trigonometric functions into a sum or difference of two other trigonometric functions.
For example, the product of two sine functions, say \(\sin(a)\) and \(\sin(b)\), can be rewritten as \(\frac{1}{2} [\cos(a-b) - \cos(a+b)]\). This identity is particularly useful in the integration problem involving the product of \(\sin(3x)\) and \(\sin(7x)\).

By substituting the specific values \(a = 3x\) and \(b = 7x\) into the identity, the product is transformed into \(\frac{1}{2} [\cos(4x) - \cos(10x)]\).

• This process simplifies the integral into a more manageable form.
• Instead of integrating a product, you now have two separate cosine terms that are relatively straightforward to integrate.

Finding antiderivatives, also known as integration, involves determining a function whose derivative is the given function. When dealing with trigonometric functions, it's useful to know specific antiderivative formulas.
The antiderivative of \(\cos(kx)\) can be expressed as \(\frac{1}{k} \sin(kx) + C\), where \(C\) is the constant of integration that accounts for there being an infinite number of antiderivatives. This is because the derivative of a constant is zero.

In this solution, each cosine term obtained from the product-to-sum conversion is integrated separately:

• For \(\frac{1}{2} \int \cos(4x) \, dx\), you would use \(\frac{1}{4} \sin(4x) + C\) as the antiderivative.
• For \(\frac{1}{2} \int \cos(10x) \, dx\), you have \(\frac{1}{10} \sin(10x) + C\) as the antiderivative.

After finding these, you combine them, ensuring that the constants are included in the final simplified solution as \(\frac{1}{8} \sin(4x) - \frac{1}{20} \sin(10x) + C\).

• This process is fundamental in many areas of calculus, allowing the reversal of the differentiation process.