Reviewed by Laila A. Lico
Find the integral of 1/x?
Answer: The integral of 1/x is ln|x| + C
The integral of 1/x is a fundamental result in calculus, serving as a classic example of an antiderivative involving logarithms. Integrating rational functions like 1/x is common in both elementary and advanced mathematics, and understanding this process helps build a solid foundation for further studies in calculus and analysis.
Methods
Math Tutor Explanation Using Basic Integration Rules
This approach uses the fact that the derivative of the natural logarithm of the absolute value of x is 1/x.
Step 1: Step 1: Recall that d/dx[ln|x|] = 1/x
Step 2: Step 2: Thus, the antiderivative (integral) of 1/x is ln|x| + C
Math Tutor Explanation Using Substitution Method
This method can reaffirm the answer by making a substitution, even though it is direct, to encourage understanding the process.
Step 1: Step 1: Set u = x, then du = dx
Step 2: Step 2: Integrate 1/u du, which gives ln|u| + C
Step 1:
Step 2:
Math Tutor suggests: Explore More Integral and Derivative Problems
Boost your understanding of calculus by practicing related integral and derivative exercises.
FAQ on Integrals Involving 1/x
Why do we use absolute value in ln|x|?
Because the natural logarithm is only defined for positive numbers, but 1/x is defined for x ≠ 0. Using ln|x| ensures the result is valid for both positive and negative x (except zero).
What is the constant of integration (C)?
The constant of integration represents any arbitrary constant added to the indefinite integral, since differentiation of a constant is zero.
Can you integrate 1/x^2 the same way?
No, integrating 1/x^2 uses the power rule and results in -1/x + C.
What does the integral of 1/x represent graphically?
It represents the area under the curve y=1/x, which accumulates as the natural logarithm of the absolute value of x.
How do you evaluate the definite integral of 1/x from a to b?
Calculate ln|b| - ln|a|, which simplifies to ln(|b|/|a|), as long as both a and b are nonzero and have the same sign.
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