Introduction

Integration is a fundamental operation in calculus, used to find areas, volumes, and solve problems in various fields such as physics, engineering, and economics. In this article, we will delve into the process of solving the definite integral of x2 with respect to x, and the indefinite integral of secy with respect to dy. We will explore the step-by-step procedures and present the solutions with clarity and precision.

Solving the Definite Integral of x2

The definite integral of x2 with respect to x can be written as:

∫ab x2 dx

To solve this, we integrate x2 using the power rule for integration, which states that ∫x^n dx xn 1 / (n 1) C, where C is the constant of integration. Here, n 2, thus:

∫ x2 dx x3 / 3 C

Evaluating this at the limits a and b, we get:

[b3 / 3] - [a3 / 3] b3 / 3 - a3 / 3

This is the solution for the definite integral of x2 from a to b.

Solving the Indefinite Integral of secy

The indefinite integral of secy with respect to dy can be written as:

∫ secy dy

This integral is more complex and requires a different approach. We can use the integral of the secant function, which is:

∫ secy dy ln |secy tany| C

Here, C is the constant of integration. The proof of this formula involves the substitution method and the trigonometric identity for secy and tany.

Combining the Solutions

Now, we consider the equations provided in the article:

∫ x2 dx x3 / 3 C

and:

∫ secy dy ln |secy tany| C

By equating the two integrals as presented in the article, we get:

x3 / 3 ln |secy tany| C

Solving for x3, we get:

x3 3 ln |secy tany| 3C

This is the combined solution of the two integrals as per the given equation.

Conclusion

In conclusion, the process of solving integrals, whether definite or indefinite, involves recognizing the appropriate integration techniques and applying them accurately. The solutions presented here demonstrate the power of calculus in solving complex mathematical problems. The key to mastering these techniques lies in practice and understanding the underlying principles.

References and Further Reading

1. Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.

2. Larson, R., Hostetler, R. (2018). Calculus of a Single Variable. Cengage Learning.

3. Rex Lanham. (2023). Advanced Techniques in Calculus. Cambridge University Press.