Question: Evaluate the integral: ∫ (1/(x^2 + 1)) dx
Options:
tan^(-1)(x) + C
sin^(-1)(x) + C
ln
x
+ C
cos^(-1)(x) + C
Correct Answer: tan^(-1)(x) + C
Solution:
The integral of 1/(x^2 + 1) is tan^(-1)(x) + C.
Evaluate the integral: ∫ (1/(x^2 + 1)) dx
Practice Questions
Q1
Evaluate the integral: ∫ (1/(x^2 + 1)) dx
tan^(-1)(x) + C
sin^(-1)(x) + C
ln
x
Questions & Step-by-Step Solutions
Evaluate the integral: ∫ (1/(x^2 + 1)) dx
Correct Answer: tan^(-1)(x) + C
Step 1: Identify the integral you need to evaluate, which is ∫ (1/(x^2 + 1)) dx.
Step 2: Recognize that the function 1/(x^2 + 1) is a standard integral that corresponds to the derivative of the arctangent function.
Step 3: Recall that the derivative of tan^(-1)(x) is 1/(x^2 + 1).
Step 4: Therefore, the integral ∫ (1/(x^2 + 1)) dx equals tan^(-1)(x) plus a constant of integration, C.
Step 5: Write the final answer as tan^(-1)(x) + C.
Integration of Rational Functions – This concept involves finding the antiderivative of functions that can be expressed as a ratio of polynomials, particularly recognizing forms that correspond to standard integral results.
Inverse Trigonometric Functions – Understanding the relationship between certain integrals and their corresponding inverse trigonometric functions, such as the integral of 1/(x^2 + 1) being related to arctan.
Soulshift Feedback×
On a scale of 0–10, how likely are you to recommend
The Soulshift Academy?
