The Antiderivative of a sinx b cosx2 and Its Applications

In the field of calculus, understanding the antiderivative of a function is fundamental to solving a wide range of problems. This article will focus on the antiderivative of the function a sinx b cosx2, which can be decomposed using trigonometric identities and integrated step-by-step. We will also explore the practical applications of this calculation.

Introduction to Integral Calculus

Integral calculus deals with the process of finding integrals, which measure the accumulation of a quantity. The antiderivative, or integral, of a function can be seen as the reverse operation of differentiation. For a given function f(x), the antiderivative is a function F(x) such that F'(x) f(x).

Decomposing the Function: a sinx b cosx2

Consider the function I ∫a sinx b cosx2 dx. To find its antiderivative, we need to decompose and integrate each term separately.

Step-by-Step Solution

Let's rewrite the function as follows:

I ∫(a sinx b cosx2) dx

We can split this integral into two parts:

I ∫a sinx dx ∫b cosx2 dx

Let's integrate each term separately:

∫a sinx dx is a straightforward integral. The antiderivative of sinx is -cosx. Therefore, we have:

∫a sinx dx -a cosx C1

∫b cosx2 dx requires more attention. We can use a trigonometric identity to simplify the cosine term.

cos2x (1 cos2x) / 2

Thus, we can rewrite the integral as:

∫b cosx2 dx ∫b (1 cos2x) / 2 dx

This can be further split into:

∫b (1 cos2x) / 2 dx (b/2) ∫1 dx (b/2) ∫cos2x dx

The first integral is simply:

(b/2) ∫1 dx (b/2)x C2

The second integral requires a substitution. Let u 2x, so du 2 dx. Therefore:

(b/2) ∫cos2x dx (b/4) ∫cosu du (b/4) sinu C3 (b/4) sin2x C3

Combining all parts, we get:

∫b cosx2 dx (b/2)x (b/4) sin2x C2

Finding the combined antiderivative:

I -a cosx (b/2)x (b/4) sin2x C

Where C is the constant of integration.

Practical Applications

The antiderivative of a sinx b cosx2 has several practical applications in physics, engineering, and mathematics. Some examples include:

Vibrations and Waves: Understanding the behavior of oscillating systems, such as springs or pendulums. Signal Processing: Analyzing and modifying signals, such as sound and electrical signals. Mechanical Systems: Modeling and predicting the motion of objects in various mechanical systems.

Conclusion

Understanding and calculating the antiderivative of functions like a sinx b cosx2 is a crucial skill in calculus. This process involves breaking down the function into simpler components, applying trigonometric identities, and integrating each term separately. The applications of such calculations are broad and range from theoretical mathematics to practical engineering problems.

If you have any further questions or need more detailed explanations, feel free to reach out!

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