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Chapter 5: Problem 13

Evaluate the indefinite integrals by using the given substitutions to reduce the integrals to standard form. $$\int \sqrt{x} \sin ^{2}\left(x^{3 / 2}-1\right) d x, \quad u=x^{3 / 2}-1$$

Short Answer

Expert verified
The indefinite integral evaluates to \( \frac{1}{3} (x^{3/2} - 1 - \frac{1}{2} \sin(2(x^{3/2} - 1))) + C \).

Step by step solution

01

Identify the Substitution

We are given the substitution \( u = x^{3/2} - 1 \) for the integral \( \int \sqrt{x} \sin^2(x^{3/2} - 1) \, dx \). Our goal is to express \( dx \) and other terms in terms of \( du \) and \( u \).
02

Differentiate the Substitution

Differentiate \( u \) with respect to \( x \):\[ \frac{du}{dx} = \frac{d}{dx}(x^{3/2} - 1) = \frac{3}{2}x^{1/2} \]Thus, we have \( du = \frac{3}{2}x^{1/2} \, dx \). Solve for \( x^{1/2} \, dx \):\[ x^{1/2} \, dx = \frac{2}{3} \, du \].
03

Express the Integral in Terms of u

Substitute the expression for \( x^{1/2} \, dx \) into the integral:\[ \int \sqrt{x} \sin^{2}(x^{3/2} - 1) \, dx = \int \sin^2(u) \cdot \frac{2}{3} \, du \].This simplifies to:\[ \frac{2}{3} \int \sin^2(u) \, du \].
04

Use the Trigonometric Identity

We use the identity \( \sin^2(u) = \frac{1 - \cos(2u)}{2} \) to simplify the integral:\[ \frac{2}{3} \int \left(\frac{1 - \cos(2u)}{2}\right) \, du = \frac{1}{3} \int (1 - \cos(2u)) \, du \].
05

Integrate the Simplified Expression

Integrate each term separately:\[ \int 1 \, du = u \]\[ \int \cos(2u) \, du = \frac{1}{2}\sin(2u) \]Substitute these back into the integral:\[ \frac{1}{3} \left( u - \frac{1}{2} \sin(2u) \right) + C \], where \( C \) is the constant of integration.
06

Substitute Back for x

Recall \( u = x^{3/2} - 1 \). Substitute \( u \) back into the integrated expression:\[ \frac{1}{3} \left( x^{3/2} - 1 - \frac{1}{2} \sin(2(x^{3/2} - 1)) \right) + C \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration by Substitution
Integration by substitution is a useful technique for solving indefinite integrals. This method transforms a complex integral into a simpler form. Here's how it works: you choose a substitution鈥攗sually a part of the integrand鈥攁nd substitute it with a new variable. In the exercise given, the substitution is denoted by \( u = x^{3/2} - 1 \). This makes the integral easier to evaluate.

  • First, you identify an expression within the integral to use as the substitution. Here, \( x^{3/2} - 1 \) is replaced by \( u \).
  • Next, you compute the derivative of \( u \) with respect to \( x \), resulting in \( \frac{du}{dx} = \frac{3}{2}x^{1/2} \).
  • This derivative is then used to express \( dx \) in terms of \( du \): \( x^{1/2} \, dx = \frac{2}{3} \, du \).
By substituting these terms, the integral simplifies to a form that is usually much easier to handle. This step is critical because it reduces the complexity of the integration task.
Trigonometric Identities
Trigonometric identities are essential tools that simplify the integration of trigonometric functions. In this exercise, after you've made the substitution, you encounter an integral involving \( \sin^2(u) \). The identity \( \sin^2(u) = \frac{1 - \cos(2u)}{2} \) is particularly useful.

  • This identity transforms the integral of \( \sin^2(u) \) into a much simpler form.
  • When applied, the integral becomes \( \frac{1}{3} \int (1 - \cos(2u)) \, du \).
  • Trigonometric identities like this are powerful because they convert complex trigonometric expressions into polynomials, which are generally straightforward to integrate.
Utilizing these identities reduces the integral to terms involving constants and simple cosine functions, facilitating easier and quicker integration by taking advantage of familiar patterns and formulas.
Differentiation
Differentiation plays a pivotal role in integration by substitution. It helps express one variable in terms of another, making the integration more accessible.

  • In the example, to change variables from \( x \) to \( u \), you compute the derivative of the substitution function \( u = x^{3/2} - 1 \).
  • The derivative, \( \frac{du}{dx} = \frac{3}{2}x^{1/2} \), allows you to express the original \( dx \) part of the integral as a function of \( du \).
  • This transformation is vital as it translates the integral from one variable to another, opening up new pathways to solve it.
Successfully employing differentiation in integration by substitution is like finding the right key to a lock. It helps bridge the gap between an initially complex expression and a manageable one, leading to the solution of the indefinite integral.

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