Integrating (16x - 15)^18 Demystifying The Power Rule And U-Substitution

In the realm of calculus, integration stands as a fundamental operation, representing the reverse process of differentiation. Mastering integration techniques is crucial for solving a wide array of problems in mathematics, physics, engineering, and various other scientific disciplines. When confronted with an indefinite integral, the initial step involves identifying the appropriate integration formula that aligns with the structure of the integrand. This article delves into the process of selecting the basic integration formula applicable to the indefinite integral ∫(16x - 15)^18 dx, offering a detailed explanation and guiding you through the solution.

Understanding Indefinite Integrals

Before we dive into the specific integral at hand, it's essential to grasp the concept of indefinite integrals. An indefinite integral represents the family of functions whose derivative is equal to the given integrand. The process of finding an indefinite integral is known as antidifferentiation. The result of indefinite integration always includes a constant of integration, denoted by 'C', which accounts for the fact that the derivative of a constant is zero.

The general form of an indefinite integral is expressed as:

∫f(x) dx = F(x) + C

Where:

• f(x) is the integrand (the function to be integrated).
• F(x) is the antiderivative of f(x).
• C is the constant of integration.

Identifying the Appropriate Integration Formula

When faced with an indefinite integral, the key lies in recognizing the pattern of the integrand and matching it with a suitable basic integration formula. There are several fundamental integration formulas, each applicable to specific types of functions. Some common integration formulas include:

1. Power Rule: ∫x^n dx = (x^(n+1))/(n+1) + C (where n ≠ -1)
2. Integral of 1/x: ∫(1/x) dx = ln|x| + C
3. Integral of e^x: ∫e^x dx = e^x + C
4. Integral of a^x: ∫a^x dx = (a^x)/ln(a) + C
5. Integrals of Trigonometric Functions:
   • ∫sin(x) dx = -cos(x) + C
   • ∫cos(x) dx = sin(x) + C
6. Integrals of Inverse Trigonometric Functions: (e.g., ∫(1/√(1-x^2)) dx = arcsin(x) + C)

Analyzing the Integrand: ∫(16x - 15)^18 dx

Now, let's focus on the integral we aim to solve: ∫(16x - 15)^18 dx. The integrand is (16x - 15)^18. Observe that this expression is a composite function, where a linear function (16x - 15) is raised to a power (18). This structure suggests that the u-substitution method, combined with the power rule, would be an effective approach. The power rule, in its general form, is expressed as:

∫u^n du = (u^(n+1))/(n+1) + C

Where:

• u is a function of x.
• n is a constant (n ≠ -1).

Applying u-Substitution and the Power Rule

To apply the u-substitution method, we need to identify a suitable 'u' within the integrand. In this case, a natural choice for 'u' is the linear function inside the parentheses:

u = 16x - 15

Next, we find the derivative of 'u' with respect to 'x':

du/dx = 16

Now, we solve for 'dx':

dx = du/16

We can now substitute 'u' and 'dx' back into the original integral:

∫(16x - 15)^18 dx = ∫u^18 (du/16)

We can pull the constant 1/16 out of the integral:

(1/16) ∫u^18 du

Now, we can apply the power rule:

(1/16) * (u^19)/19 + C

Finally, we substitute back the original expression for 'u':

(1/16) * ((16x - 15)^19)/19 + C

Simplifying, we get:

((16x - 15)^19)/304 + C

Why Other Options Are Not Suitable

Let's briefly discuss why the other options provided (∫e^u du and ∫du/u) are not the primary choices for this particular integral:

• ∫e^u du: This formula is applicable when the integrand involves an exponential function with a variable exponent. Our integrand is a polynomial function raised to a power, not an exponential function.
• ∫du/u: This formula is used when the integrand is a fraction where the numerator is the derivative of the denominator. While u-substitution might lead to a fraction, the initial form of the integrand doesn't directly fit this pattern.

Conclusion

In summary, the basic integration formula that is most suitable for solving the indefinite integral ∫(16x - 15)^18 dx is ∫u^n du, which represents the power rule in its general form. By employing the u-substitution method, we effectively transformed the integral into a form where the power rule could be readily applied. This process demonstrates the importance of recognizing patterns within integrands and selecting the appropriate integration techniques to arrive at the solution. Mastering these fundamental integration techniques is crucial for success in calculus and its applications.

Integration, a cornerstone of calculus, is the reverse operation of differentiation. It involves finding the antiderivative of a function. In simpler terms, if we know the rate of change of a quantity, integration helps us find the original quantity. Mastering integration is crucial for solving problems in various fields, including physics, engineering, economics, and computer science. This article focuses on two essential techniques: the power rule and u-substitution, which are fundamental for tackling a wide range of integration problems. We'll specifically explore how these techniques apply to the indefinite integral ∫(16x - 15)^18 dx, providing a step-by-step guide to its solution.

The Essence of Integration: Reversing Differentiation

To truly grasp integration, it's vital to understand its connection to differentiation. Differentiation allows us to find the derivative of a function, which represents the instantaneous rate of change. Integration, on the other hand, reverses this process. It allows us to find a function whose derivative is a given function. This