Problem 10

Question

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int\left(1-\cos \frac{t}{2}\right)^{2} \sin \frac{t}{2} d t, \quad u=1-\cos \frac{t}{2}$$

Step-by-Step Solution

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Answer

The evaluated integral is $ \frac{2}{3}(1 - \cos \frac{t}{2})^3 + C $.

1Step 1: Apply the Substitution

Given the substitution $ u = 1 - \cos \frac{t}{2} $, we first need to find the derivative of $ u $ with respect to $ t $. Differentiate to get $ du = \frac{1}{2} \sin \frac{t}{2} \, dt $, which rearranges to $ dt = 2 \frac{du}{\sin \frac{t}{2}} $.

2Step 2: Rewrite the Integral

Substitute $ u = 1 - \cos \frac{t}{2} $ and $ dt = 2 \frac{du}{\sin \frac{t}{2}} $ into the integral. The integral becomes $ \int u^2 \sin \frac{t}{2} \cdot 2 \frac{du}{\sin \frac{t}{2}} $, simplifying to $ 2 \int u^2 \, du $.

3Step 3: Integrate in Terms of u

The integral $ 2 \int u^2 \, du $ is a standard power of $ u $, which integrates to $ 2 \cdot \frac{u^3}{3} + C $. This simplifies to $ \frac{2}{3}u^3 + C $.

4Step 4: Substitute Back to Original Variable

Replace $ u $ with the original expression, $ u = 1 - \cos \frac{t}{2} $. The solution in terms of $ t $ is $ \frac{2}{3}(1 - \cos \frac{t}{2})^3 + C $.

Key Concepts

Understanding the Substitution Method in IntegrationSimplifying Integration with Trigonometric IntegralsCalculus Integration Techniques for Indefinite Integrals

Understanding the Substitution Method in Integration

The substitution method is a powerful tool in calculus for evaluating integrals. It's especially helpful for transforming complex integrals into simpler forms that are easier to solve. Here's how it works: you choose a substitution that simplifies part of the integral, often turning it into a standard form that you can integrate easily.In the context of the original exercise, the substitution was given as $ u = 1 - \cos \frac{t}{2} $. This effectively turns a somewhat complicated integrand into a simpler polynomial function in terms of $ u $.

First, find the expression for $ du $ by differentiating $ u $ with respect to $ t $. This helps in expressing $ dt $ in terms of $ du $.
Next, substitute the expressions for $ u $ and $ dt $ back into the integral, transforming it into an easier format.

By using substitution, the original integral transforms from an involving trigonometric form to a more straightforward polynomial form $ 2 \int u^2 \, du $. This transformation is key to solving the problem efficiently.

Simplifying Integration with Trigonometric Integrals

Trigonometric integrals are often encountered in calculus, involving functions like sine, cosine, and their various powers. Solving these integrals can sometimes be tricky, but they often become more manageable through clever substitutions.In the exercise provided, the integral involves powers of trigonometric functions. The form $ \int \left(1-\cos \frac{t}{2}\right)^{2} \sin \frac{t}{2} \, dt $ initially looks complex.

The strategy involves converting the trigonometric expressions to a simpler variable $ u $, as shown in the solution. This makes use of identities or given substitutions to reduce the integral to a standard, solvable form.
It's valuable to recognize when integral expressions involving trigonometric functions can become simpler polynomials through proper substitutions. Hence, knowing common identities and substitution tricks is essential.

Through these techniques, integration becomes not just possible but straightforward, eliminating the initial complexity of trigonometric integrals.

Calculus Integration Techniques for Indefinite Integrals

Understanding various integration techniques is pivotal in solving indefinite integrals effectively. In calculus, there are several strategies beyond substitution that help tackle different forms of integrals.The step-by-step process to solve the given problem also highlights standard integration techniques:

The direct integration of power functions is a vital tool, used when the integral is expressed in terms of simple powers, as in $ \int u^2 \, du $.
After simplifying the integral using substitution, integrating a polynomial function involves applying the power rule: $ \int u^n \, du = \frac{u^{n+1}}{n+1} + C $ where $ C $ is the constant of integration.
Subsequently, back-substituting the original variable is crucial to align the answer with the context of the problem.

These techniques form the backbone of calculus and enable students to tackle a wide range of indefinite integrals systematically and confidently. Mastery of these methods ensures a solid foundation for progressing in mathematics.

Problem 10

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