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

2:58 minutes

Problem 8.1.38

Textbook Question

7–64. Integration review Evaluate the following integrals.
38. ∫ x / (x⁴ + 2x² + 1) dx

Verified step by step guidance

Step 1: Observe the integrand ∫ x / (x⁴ + 2x² + 1) dx. Notice that the denominator can be factored. Rewrite the denominator as (x² + 1)², since x⁴ + 2x² + 1 is a perfect square trinomial.

Step 2: Substitute u = x² + 1 to simplify the integral. Compute the derivative of u with respect to x: du/dx = 2x, which implies du = 2x dx.

Step 3: Rewrite the integral in terms of u. Substitute x dx with (1/2) du, and the denominator becomes u². The integral now becomes (1/2) ∫ 1/u² du.

Step 4: Apply the power rule for integration to ∫ 1/u² du. Recall that ∫ u⁻² du = -u⁻¹ + C, where C is the constant of integration.

Step 5: Substitute back u = x² + 1 into the result to express the solution in terms of x. The final answer will be in the form of -1/(x² + 1) + C.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

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 find the integral of a function. Common techniques include substitution, integration by parts, and partial fraction decomposition. Understanding these methods is crucial for evaluating more complex integrals, such as the one presented in the question.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers. In the integral ∫ x / (x⁴ + 2x² + 1) dx, the denominator is a polynomial of degree four. Recognizing the structure of polynomial functions helps in simplifying the integral and determining appropriate integration techniques.

Rational Functions

Rational functions are ratios of two polynomial functions. The integral in the question involves a rational function, which can often be simplified or decomposed for easier integration. Understanding how to manipulate rational functions is essential for effectively evaluating integrals like the one given.