Integrating Functions Using Substitution Techniques: A Comprehensive Guide

In calculus, the technique of substitution is a powerful method for evaluating integrals. This guide will walk you through the process of integrating a specific function using the substitution method. We will explore the integration of the function 1/2x ? 16 by introducing the substitution variable u. Through detailed steps and calculations, we aim to demonstrate the step-by-step process and the application of the substitution method in integral calculus.

Introduction to the Substitution Method

The substitution method, or u-substitution, is a fundamental technique used in integral calculus. It involves re-expressing the integral in terms of a new variable, typically denoted as u, to simplify the integration process. The general form of a u-substitution integral is as follows:

I ∫ f(u) du

Where u is a function of x, and we set du g(x) dx. The key step is to rewrite the original function in terms of u and du, making the integral easier to solve.

Integrating (x(2x - 1)^6) Using Substitution

Consider the integral I ∫ 1/2x(2x - 1)6 dx. To solve this integral using substitution, we introduce the substitution u 2x - 1. This substitution will facilitate the simplification of the function and make the integration process more manageable.

Step 1: Introduction of the Substitution

Define the substitution:

u 2x - 1 x 1/2u 1

And compute the differential:

du 2 dx dx 1/2du

Step 2: Rewrite the Integral in Terms of u

Substitute x and dx in the original integral:

∫ 1/2x(2x - 1)6 dx ∫ 1/2(u 1) u6 (1/2du)

Simplify the expression:

∫ 1/2(u 1) u6 du ∫1/4(u7 u6) du

Step 3: Integrating with Respect to u

Now, integrate the function with respect to u:

∫1/4(u7 u6) du 1/4(1/8u8 1/7u7) C

Step 4: Substituting Back to x

Finally, substitute back for u to get the integral in terms of x:

1/4(1/8u8 1/7u7) C 1/32u8 1/28u7 C

1/224u7(7u 14) C

1/224(2x - 1)7(14x - 1) C

Alternative Solution without Substitution

Another approach to solve the same integral is to avoid the substitution and directly manipulate the function:

∫ 1/2x(2x - 1)6 dx

Start by separating the integral:

1/2 ∫ 2x(2x - 1)6 dx

Note that 2x can be written as u - 1, leading to the equivalent form:

1/2 ∫ (u - 1)6 du

Expand and integrate term by term:

1/32(2x - 1)8 - 1/28(2x - 1)7 C

By arranging the terms:

1/224(2x - 1)7(14x - 1) C

Conclusion

The process of integrating x(2x - 1)6 using the substitution method or the alternative approach showcases the power and flexibility of substitution in solving complex integrals. Whether you choose to use substitution or follow a more direct path, the result is the same. Understanding these techniques is crucial for mastering advanced calculus and solving a wide range of integration problems.