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Chapter 8: Problem 40

The integrals in Exercises \(1-44\) are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate. When necessary, use a substitution to reduce it to a standard form. $$ \begin{array}{l}{\int \frac{\sqrt{x}}{1+x^{3}} d x} \\ {\text {Hint: Let } u=x^{3 / 2}}\end{array} $$

Short Answer

Expert verified
\( \frac{2}{3} \arctan(x^{3/2}) + C \)

Step by step solution

01

Substitution

Use the hint provided to let \( u = x^{3/2} \). Then, differentiate \( u \) with respect to \( x \) to find \( \frac{du}{dx} \), giving \( du = \frac{3}{2}x^{1/2} \, dx \). This implies \( dx = \frac{2}{3x^{1/2}} \, du \).
02

Simplify the Integral

Substitute \( u = x^{3/2} \) in the integral \( \int \frac{\sqrt{x}}{1+x^3} \, dx \). Replace \( dx \) using the expression derived: \( dx = \frac{2}{3x^{1/2}} \, du \). The integrand becomes \( \int \frac{\sqrt{x}}{1+u^2} \cdot \frac{2}{3x^{1/2}} \, du \).
03

Cancel and Reintegrate

Now, observe that \( \sqrt{x} \) and \( x^{1/2} \) cancel out. Thus, the integral simplifies to \( \frac{2}{3} \int \frac{1}{1+u^2} \, du \), which is a standard form.
04

Evaluate the Standard Integral

Recognize that \( \int \frac{1}{1+u^2} \, du \) is the arctangent function. Thus, \( \frac{2}{3} \int \frac{1}{1+u^2} \, du = \frac{2}{3} \arctan(u) + C \), where \( C \) is the constant of integration.
05

Back-Substitute for x

Substitute back \( u = x^{3/2} \) to get the final answer in terms of \( x \). Therefore, the integral evaluates to \( \frac{2}{3} \arctan(x^{3/2}) + C \).

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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 vital technique in calculus, especially for solving integrals that are not readily solvable in their original form. This method involves changing variables to simplify an integral into a form that is easier to work with. Here's how it works:
  • Identify a part of the integral that can be substituted with a single variable, often denoted as \( u \).
  • Differentiate this substitution with respect to \( x \) to find \( \frac{du}{dx} \), then solve for \( dx \).
  • Replace both the identified part of the integrand and the differential \( dx \) with the expressions in terms of \( u \) and \( du \).
  • This should convert the integral into a simpler form, often reducing it to a standard integral.
  • Once the integral is evaluated in terms of \( u \), substitute back the original variable to express the answer in terms of \( x \).
In the example provided, by letting \( u = x^{3/2} \), the given integral was transformed into a standard arctan form, making it much easier to solve.
Trigonometric Integrals
Trigonometric integrals involve algebraic expressions with trigonometric functions and often need specific techniques to simplify and solve. These integrals typically use identities or substitutions to rewrite and evaluate them.For instance:
  • Recognize patterns within the integrand that match with trigonometric identities, which can simplify the expression.
  • When direct trigonometric identities do not apply, employ substitution to relate the trigonometric part of the integrand to a simpler expression.
  • Integrals can sometimes reduce into standard forms that are easier to handle, such as \( \int \frac{1}{1+u^2} \, du \), which is known to be \( \arctan(u) \).
Although the given exercise did not explicitly involve trigonometric functions at the start, it skillfully reduced the problem to a standard trigonometric integral using substitution.
Calculus Exercises
Calculus exercises enhance understanding and proficiency in applying calculus concepts like integration. Tackling different types of problems, such as those involving algebraic, logarithmic, exponential, or trigonometric expressions, strengthens mathematical flexibility and skills. In approaching exercises:
  • Analyze the problem to decide which method (algebraic manipulation, substitution, or trigonometric identity) will be the most efficient.
  • Break down the exercise into manageable steps to ensure each part of the integral is being treated correctly.
  • Keep practicing, as solving various integrals consolidates knowledge of calculus techniques and methods.
By regularly engaging with diverse calculus exercises, one can grasp the necessity of techniques such as substitution and build confidence in using them to solve complex integrals effectively.

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Most popular questions from this chapter

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