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Magazine Chapter 5: Problem 52
If you do not know what substitution to make, try reducing the integral step by step, using a trial substitution to simplify the integral a bit and then another to simplify it some more. You will see what we mean if you try the sequences of substitutions. \(\int \sqrt{1+\sin ^{2}(x-1)} \sin (x-1) \cos (x-1) d x\) a. \(u=x-1,\) followed by \(v=\sin u,\) then by \(w=1+v^{2}\) b. \(u=\sin (x-1),\) followed by \(v=1+u^{2}\) c. \(u=1+\sin ^{2}(x-1)\)
Short Answer
Expert verified
All substitutions lead to \( \int \sqrt{1 + \sin^2(x-1)} \sin(x-1) \cos(x-1) \, dx = \frac{1}{3} (1 + \sin^2(x-1))^{3/2} + C \).
Step by step solution
01
Initial Setup for Approach a
We start by using the substitution \( u = x - 1 \). This gives us \( du = dx \).The integral now becomes \( \int \sqrt{1 + \sin^2(u)} \sin(u) \cos(u) \, du \).
02
Substitute with Approach a - v-substitution
Next, we use the substitution \( v = \sin(u) \), so \( dv = \cos(u) \, du \).The integral can be rewritten as \( \int \sqrt{1 + v^2} \, v \, dv \).
03
Simplify with Approach a - w-substitution
Now use \( w = 1 + v^2 \), giving \( dw = 2v \, dv \) or \( dv = \frac{dw}{2v} \).Substituting \( v \, dv \) gives the integral \( \frac{1}{2} \int \sqrt{w} \, dw \).
04
Solve the Integral for Approach a
The integral \( \frac{1}{2} \int \sqrt{w} \, dw \) is now straightforward.This simplifies to \( \frac{1}{2} \times \frac{2}{3} w^{3/2} + C = \frac{1}{3} (1 + v^2)^{3/2} + C \).Re-substitute \( v = \sin(u) \) to get \( \frac{1}{3} (1 + \sin^2(u))^{3/2} + C \).Finally, replace \( u = x - 1 \), resulting in \( \frac{1}{3} (1 + \sin^2(x-1))^{3/2} + C \).
05
Initial Setup for Approach b
Let's try the substitution \( u = \sin(x-1) \), so \( du = \cos(x-1) \, dx \).Our integral becomes \( \int \sqrt{1 + u^2} \, u \, du \).
06
Simplify and Solve with Approach b
The integral is exactly \( \int \sqrt{1 + u^2} \, u \, du \), which we've encountered.Following the same steps as before gives \( \frac{1}{3} (1 + u^2)^{3/2} + C \).Substituting back \( u = \sin(x-1) \), we get \( \frac{1}{3} (1 + \sin^2(x-1))^{3/2} + C \).
07
Direct Setup for Approach c
For this approach, substitute \( u = 1 + \sin^2(x-1) \), so \( du = 2\sin(x-1) \cos(x-1) \, dx \).This simplifies to \( \int \sqrt{u} \frac{1}{2} \, du = \frac{1}{2} \int \sqrt{u} \, du \).
08
Solve the Integral for Approach c
This is a direct integration problem.It results in \( \frac{1}{2} \times \frac{2}{3} u^{3/2} + C = \frac{1}{3} u^{3/2} + C \).Since \( u = 1 + \sin^2(x-1) \), write the final answer as \( \frac{1}{3} (1 + \sin^2(x-1))^{3/2} + C \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method is a powerful tool in integration, often used to simplify complex integrals. The primary idea is to replace a complicated part of the integral with a single variable, making it easier to solve. In the provided exercise, we see three approaches using substitution, each aiming to simplify the integral step by step.
- Start with an appropriate substitution to transform the integral into a simpler form. For instance, substituting a trigonometric expression to linearize the structure.
- Continue simplifying the integral with subsequent substitutions, if needed. As demonstrated, multiple layers of substitution can break down even complex integrals.
- Finally, resolve the simple integral and back-substitute to revert to the original variable before substitution.
Trigonometric Integration
Trigonometric integration involves integrating expressions with trigonometric functions like sine and cosine. These types of integrals often benefit from substitution, especially when the integral has composite trigonometric forms, as seen in the exercise. Let's break it down:
- Use trigonometric identities to simplify expressions. For example, knowing that \( \sin^2(u) \) and \( \cos^2(u) \) can be related by the identity \( \sin^2(u) + \cos^2(u) = 1 \), allows you to substitute one in terms of the other.
- Transforming a complex argument like \( \sin(x-1) \) via substitution makes it easier to handle, converting it to simpler variables like \( v = \sin(u) \).
- Trigonometric integrals are greatly simplified using appropriate substitutions, making the integral more manageable by reducing complexity through identity recognition.
Step-by-Step Integration
Step-by-step integration involves systematically breaking down the integration process into manageable steps. This structured method helps in clarifying the approach while solving integrals, especially for complex ones.
- Begin with identifying a substitution that simplifies the variable or the function within the integral. Each step should contribute towards making the integral easier.
- In the exercise, we see multiple steps where each substitution is carefully chosen to simplify the problem incrementally, leading towards a straightforward integral.
- Solve each simplified integral step by step. Inserting appropriate substitutions at each stage ensures clarity and streamlines the problem-solving process.
- After solving, retrace the steps in reverse order to substitute back the original variables.
Integral Simplification
Integral simplification focuses on reducing the complexity of an integral to make it easier to solve. It often involves a combination of algebraic manipulation and substitution.
- Simplification can start with renaming variables or expressions to reduce computational complexity. This is evident when factors like \( w = 1 + v^2 \) are introduced, simplifying the radical expression in the integral.
- Identify patterns or structures in the integral. Recognizing terms that can pair up or simplify into familiar forms allows for easier integration.
- Use basic integral rules and recalibrate the integral structure, such as transforming a trigonometric-radical expression into a simple polynomial integral.
- Throughout simplification, keep the ultimate goal in mind: obtaining a simpler form that can easily be integrated.
