Textbook Question
92–98. Evaluate the following integrals.
94. ∫ (dt / (t³ + 1))
5
views
7. Antiderivatives & Indefinite Integrals
Indefinite Integrals
1:17 minutesProblem 4.R.101
Textbook Question
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ (x² / (x⁴ + x²)) dx
Verified step by step guidance1
Step 1: Simplify the integrand. Notice that the denominator can be factored as x²(x² + 1). Rewrite the integrand as (x² / (x²(x² + 1))) = 1 / (x² + 1).
Step 2: Recognize that the simplified integrand, 1 / (x² + 1), resembles the derivative of the arctangent function. Recall that the derivative of arctan(x) is 1 / (1 + x²).
Step 3: Adjust the integrand to match the standard form. In this case, the integrand is already in the form 1 / (x² + 1), which corresponds directly to the derivative of arctan(x).
Step 4: Integrate the simplified function. The integral of 1 / (x² + 1) is arctan(x) + C, where C is the constant of integration.
Step 5: Write the final expression for the indefinite integral as ∫ (x² / (x⁴ + x²)) dx = arctan(x) + C.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
1mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Indefinite Integrals
Indefinite integrals represent a family of functions whose derivative is the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. The process of finding an indefinite integral is often referred to as antidifferentiation, and it is fundamental in calculus for solving problems related to area under curves and accumulation functions.
Recommended video:
05:04
Introduction to Indefinite Integrals
Integration Techniques
Various techniques are employed to solve integrals, especially when dealing with complex functions. Common methods include substitution, integration by parts, and partial fraction decomposition. For the given integral, recognizing the structure of the integrand can guide the choice of technique, such as simplifying the expression or breaking it into simpler fractions for easier integration.
Recommended video:
06:18Integration by Parts for Definite Integrals
Rational Functions
A rational function is a ratio of two polynomials. In the context of integration, understanding the behavior of rational functions is crucial, particularly in simplifying the integrand. The integral provided involves a rational function, and techniques like partial fraction decomposition can be used to express it in a form that is easier to integrate, allowing for a clearer path to the solution.
Recommended video:
6:04Intro to Rational Functions
5:04mWatch next
Master Introduction to Indefinite Integrals with a bite sized video explanation from Patrick
Start learningRelated Videos
Related Practice