Evaluating Integrals Involving Sine and Cosine Functions

Integrating functions that include the sine and cosine functions is a fundamental skill in calculus, essential for solving a myriad of problems in physics, engineering, and mathematics. This article will guide you through the process of evaluating integrals involving sine and cosine functions, specifically focusing on the integral ( ∫(4sin x - 3cos x),dx ).

Understanding the Integral Operator

The symbol ( ∫ ) represents the integral operator. Integrals are used to find the area under a curve, which can be thought of as the reverse process of differentiation. The integral of a function ( f(x) ) with respect to ( x ) is denoted as ( ∫ f(x),dx ).

Step-by-Step Solution for the Given Integral

To evaluate the integral ( ∫(4sin x - 3cos x),dx ), we will follow a systematic approach.

Step 1: Distribute the Integral Over Sum or Difference

The first step is to distribute the integral over the sum or difference inside the integral. The integral of a sum or difference is the sum or difference of the integrals.

( ∫(4sin x - 3cos x),dx ∫4sin x,dx - ∫3cos x,dx )

Step 2: Factor Out Constants

The next step is to factor out any constants that are multiplying the integrand. This is done by taking the constant outside the integral.

( ∫4sin x,dx - ∫3cos x,dx 4∫sin x,dx - 3∫cos x,dx )

Step 3: Integrate Each Term Separately

Now, we need to integrate each term separately. The integrals of ( sin x ) and ( cos x ) are well-known:

( ∫sin x,dx -cos x C_1 )

( ∫cos x,dx sin x C_2 )

Step 4: Combine the Results and Include the Constant of Integration

Finally, we combine the results of the integrals and include the constant of integration, often denoted as ( C ).

( 4∫sin x,dx - 3∫cos x,dx 4(-cos x) - 3(sin x) C )

( -4cos x - 3sin x C )

Double-Checking the Solution

To verify the solution, we can take the derivative of ( -4cos x - 3sin x C ) with respect to ( x ).

( frac{d}{dx}(-4cos x - 3sin x C) 4sin x - 3cos x )

As we see, the derivative of the function we obtained is indeed the original integrand, confirming the correctness of our solution.

General Form and Applications

Integrals involving the sine and cosine functions like ( ∫(Asin x Bcos x),dx ) are common in many fields. They are used in Fourier analysis, wave theory, and other areas of physics and engineering. Understanding how to evaluate such integrals accurately is crucial for solving complex problems.

The integral's form can be generalized as:

( ∫(Asin x Bcos x),dx -Acos x Bsin x C )

Conclusion

Integral evaluation involving sine and cosine functions is a foundational concept in calculus, with wide-ranging applications. By following systematic steps, you can confidently evaluate such integrals and verify the results. Remember to always double-check your solution by differentiating the result to ensure its correctness.

Feel free to explore more integrals and see how similar techniques can be applied to solve different types of problems. Happy integrating!