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Magazine Chapter 5: Problem 29
Evaluate the integrals. $$\int x^{1 / 2} \sin \left(x^{3 / 2}+1\right) d x$$
Short Answer
Expert verified
Substitute \( u = x^{3/2} + 1 \), leading to \( dx = \frac{2}{3} x^{-1/2} du \). Integral simplifies but is complex.
Step by step solution
01
Identify the Structure
Observe that the integral \( \int x^{1/2} \sin(x^{3/2} + 1) \, dx \) involves a composite function \( \sin(u) \), where \( u = x^{3/2} + 1 \). This suggests a substitution method will be appropriate.
02
Substitution
Let \( u = x^{3/2} + 1 \). Then \( \frac{du}{dx} = \frac{3}{2}x^{1/2} \). Solving for \( dx \), we get \( dx = \frac{2}{3} x^{-1/2} du \). Substitute \( u \) and \( dx \) into the integral.
03
Simplify the Integral
Rewrite \( x^{1/2} \) in terms of \( u \) using the substitution: from \( u = x^{3/2} + 1 \), it follows \( x^{3/2} = u - 1 \) and thus \( x^{1/2} = (u - 1)^{1/3} \). Substitute into the integral to get \( \int (u-1)^{1/3} \sin(u) \frac{2}{3} \, du \).
04
Evaluate the Integral
The integral now becomes \( \frac{2}{3} \int (u-1)^{1/3} \sin(u) \, du \). Although direct integration is complex, this setup shows the substitution has simplified the integral. At this level, recognizing it's a standard form for substitution is key, and any specific solutions or techniques would need additional context, such as fitting into a known integral form or using numerical methods.
05
Rewrite in Original Terms
Although a closed-form expression isn't easily found, conceptually, you could revert any applied technique's outcome back to the variable \( x \). Recognize the integral involves complex integrals unless computational tools or numerical methods are employed.
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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 calculus for simplifying complex integrals. It involves replacing a part of the integral with a new variable, which can make the integral easier to solve.
This approach hinges on identifying a substitution that simplifies a composite function or complicated expression.
In our exercise, we observe the integral \( \int x^{1/2} \sin(x^{3/2} + 1) \, dx \), which involves a composite function \( \sin(u) \) with \( u = x^{3/2} + 1 \).
This suggests a substitution.
This approach hinges on identifying a substitution that simplifies a composite function or complicated expression.
In our exercise, we observe the integral \( \int x^{1/2} \sin(x^{3/2} + 1) \, dx \), which involves a composite function \( \sin(u) \) with \( u = x^{3/2} + 1 \).
This suggests a substitution.
- First, recognize the inside function that calls for substitution鈥攊n this case \( u = x^{3/2} + 1 \).
- Calculate \( \frac{du}{dx} = \frac{3}{2}x^{1/2} \) to relate \( du \) back to \( dx \).
- Solve for \( dx \) to use in substitution: \( dx = \frac{2}{3} x^{-1/2} du \).
Composite Function
A composite function is a function made up of two or more functions.In calculus, it often involves substituting one function into another. The challenge and interest lie in how these functions interact.In our integral \( \int x^{1/2} \sin(x^{3/2} + 1) \, dx \), the function \( \sin(x^{3/2} + 1) \) is a composite function, where \( x^{3/2} + 1 \) is substituted into the sine function.
- Recognizing composite functions is crucial for deciding on substitution methods.
- Composite functions involve the idea of an inner and outer function, like \( y=f(g(x)) \), where \( g(x) \) is the inner function and \( f \) is the outer function.
Definite Integrals
Definite integrals compute the area under a curve within a specified interval. While our original exercise focuses on an indefinite integral, understanding definite integrals helps build a foundational concept.
Unlike indefinite integrals that yield a family of functions, definite integrals give a specific number, representing the net area between the curve and the x-axis within the bounds.
Unlike indefinite integrals that yield a family of functions, definite integrals give a specific number, representing the net area between the curve and the x-axis within the bounds.
- They require an upper and lower limit on the integral symbol.
- When evaluating, use the antiderivative to compute the difference between upper and lower bounds.
- Be mindful of potentially negative areas when calculating definite integrals.
Numerical Methods
Numerical methods are techniques used to approximate solutions for mathematical problems, especially those that are difficult or impossible to solve analytically.
In contexts where substituting does not simplify enough or involves complex integrals like our example, numerical methods become a useful tool to find an approximate solution.
In contexts where substituting does not simplify enough or involves complex integrals like our example, numerical methods become a useful tool to find an approximate solution.
- Methods like the Trapezoidal Rule and Simpson's Rule approximate the area under curves.
- Numerical integration is essential when functions do not have elementary antiderivatives.
- These methods divide the area into shape families (rectangles, trapezoids) and sum their areas.
