Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

Calculus

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

Previous problemNext problem

3:05 minutes

Problem 8.1.57

Textbook Question

7–64. Integration review Evaluate the following integrals.
57. ∫ dx / (x¹⸍² + x³⸍²)

Verified step by step guidance

Step 1: Recognize that the integral ∫ dx / (x^(1/2) + x^(3/2)) involves fractional exponents. Rewrite the denominator in terms of a common factor to simplify the expression. Factor out x^(1/2) from the denominator: x^(1/2) + x^(3/2) = x^(1/2)(1 + x).

Step 2: Substitute the simplified denominator into the integral. The integral becomes ∫ dx / (x^(1/2)(1 + x)).

Step 3: Perform a substitution to simplify the integral further. Let u = 1 + x, which implies du = dx. Also, note that x = u - 1, and x^(1/2) = (u - 1)^(1/2). Rewrite the integral in terms of u.

Step 4: After substitution, the integral becomes ∫ du / ((u - 1)^(1/2) * u). Break this into manageable parts or consider partial fraction decomposition if applicable.

Step 5: Solve the resulting integral step by step, applying appropriate integration techniques such as substitution, partial fractions, or standard integral formulas. Combine the results and simplify.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration Techniques

Integration techniques are methods used to evaluate integrals, which can include substitution, integration by parts, and partial fraction decomposition. Understanding these techniques is essential for solving complex integrals, especially when dealing with rational functions or expressions that can be simplified.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the given integral, the expression in the denominator is a polynomial, and recognizing its structure is crucial for determining the appropriate method for integration, such as factoring or simplifying the expression.

Limits of Integration

Limits of integration define the interval over which the integral is evaluated. While the given integral is indefinite, understanding how to apply limits is important for definite integrals, as it affects the final value of the integral and the interpretation of the area under the curve.