Integral X 2 Lnx

Integrating the Natural Logarithm: A Comprehensive Guide to ∫x²lnx dx

Introduction:

The integral ∫x²lnx dx represents a common yet challenging problem in calculus. Understanding how to solve it is crucial for various applications in engineering, physics, and statistics, where logarithmic functions frequently appear in modeling natural phenomena. This article will break down the solution process step-by-step, exploring the underlying techniques and their implications. We’ll address this problem using integration by parts, a fundamental technique for integrating products of functions.

1. Why Integration by Parts?

Q: Why can't we directly integrate x²lnx?

A: We can't directly integrate x²lnx using basic integration rules. The integral involves a product of two functions: x² (a power function) and lnx (a logarithmic function). Neither the power rule nor the rule for integrating logarithmic functions directly applies to this combination. This necessitates the use of a technique that handles the integration of products – integration by parts.

2. Applying Integration by Parts:

Q: How does integration by parts work in this context?

A: Integration by parts is based on the product rule for differentiation: d(uv) = u dv + v du. Rearranging, we get the integration by parts formula: ∫u dv = uv - ∫v du.

To solve ∫x²lnx dx, we strategically choose:

u = lnx: This is chosen because its derivative is simpler (du = (1/x)dx).
dv = x² dx: This leaves a readily integrable function. Integrating, we get v = (x³/3).

Substituting these into the integration by parts formula:

∫x²lnx dx = (lnx)(x³/3) - ∫(x³/3)(1/x) dx

This simplifies to:

∫x²lnx dx = (x³/3)lnx - (1/3)∫x² dx

3. Completing the Integration:

Q: How do we finish solving the integral?

A: The remaining integral, ∫x² dx, is straightforward:

∫x² dx = (x³/3) + C (where C is the constant of integration)

Substituting this back into our equation:

∫x²lnx dx = (x³/3)lnx - (1/3)(x³/3) + C

Finally, we simplify to get the solution:

∫x²lnx dx = (x³/3)lnx - (x³/9) + C

4. Real-World Applications:

Q: Where might this integral appear in real-world scenarios?

A: Integrals involving logarithmic functions frequently arise in areas like:

Probability and Statistics: Certain probability density functions (like the Weibull distribution) involve logarithmic terms. Calculating expected values or other statistical measures might require evaluating integrals similar to ∫x²lnx dx.
Engineering and Physics: Logarithmic scales are common in representing quantities like sound intensity (decibels) or earthquake magnitudes (Richter scale). Integrals involving logarithms are used in analyzing systems exhibiting logarithmic behavior. For instance, calculating the work done in compressing a gas often involves integrals with logarithmic terms.
Economics: Growth models frequently employ logarithmic functions, and calculating total growth over a period might involve evaluating similar integrals.

5. Understanding the Constant of Integration:

Q: What is the significance of the constant of integration, C?

A: The constant of integration, C, is crucial because the derivative of a constant is zero. Therefore, any indefinite integral has an infinite number of possible solutions, all differing by a constant. The value of C is determined by initial conditions or boundary conditions specific to the problem being solved. For instance, if we know the value of the integral at a particular point, we can solve for C.

Conclusion:

Integrating x²lnx requires the application of integration by parts, a powerful technique for handling integrals of products. Mastering this technique is vital for tackling more complex integrals in various fields. The solution, (x³/3)lnx - (x³/9) + C, provides a foundational understanding of integrating logarithmic functions alongside polynomial functions. Remember the constant of integration (C) is essential for representing the complete family of antiderivatives.

FAQs:

1. Can we choose u and dv differently?

Yes, but choosing u = lnx and dv = x²dx is generally the most efficient strategy. Other choices will lead to more complicated integrals.

2. What if the exponent of x was different (e.g., ∫x³lnx dx)?

The process remains the same. You’d apply integration by parts, adjusting the resulting integral accordingly.

3. How can I verify my solution?

Differentiate your solution ((x³/3)lnx - (x³/9) + C). If you get back the original integrand (x²lnx), your solution is correct.

4. Can we solve this integral using numerical methods?

Yes, if analytical integration proves difficult, numerical methods (like Simpson's rule or the trapezoidal rule) can approximate the definite integral over a specific interval.

5. Are there other techniques besides integration by parts for solving integrals involving logarithms?

While integration by parts is the most common approach for this type of integral, substitution can sometimes be used in conjunction with integration by parts, particularly for more complex examples. For example, a substitution might simplify the integrand before applying integration by parts.

Search Results:

Math 1B, lecture 8: Integration by parts - Nathan Pflueger's … To obtain the integration version of this, integrate both sides with respect to x to obtain the following, which is usually written with the arguments \(x)00 suppressed, in di erential notation.

18.01A Topic 5 x F - MIT We can use this deﬁnition of lnx to derive all the properties of lnx. This is an important example of how to derive properties from functions deﬁned as integrals.

Repeated Integration and Explicit Formula for the n-th Integral of x… In this article, we extend this study to integrals by considering two types of repetitive integrals: the repeated integral and the recurrent integral. Our aim is to obtain formulae to simplify these …

Integration that leads to logarithm functions - mathcentre.ac.uk The derivative of lnx is 1 x. As a consequence, if we reverse the process, the integral of 1 x is lnx + c. In this unit we generalise this result and see how a wide variety of integrals result in …

Mat104 Fall 2002, Improper Integrals From Old Exams While we’re here let me say that the other integral, as x runs from 2 to ∞ converges. To see this, observe that x2 lnx > x2 and thus 1/(x 2lnx) < 1/x . By comparison Z ∞ 2 dx x2 lnx converges. …

Integration by Parts - Trinity University For an integral of the form Z xA(lnx)k dx, k∈ N, A∈ R, use integration by parts with u= (lnx)k, dv = xAdx, and repeat. This also works if we replace lnx by arcsinx or arccosx, or lnx by a …

Practice Problems: Integration by Parts (Solutions) - UC Santa … Find the area between the given curves: y= x2 lnx, y= 4lnx Solution: We need to nd when the two curves intersect, so set them equal to each other: x 2 lnx= 4lnx)(x 2 4)lnx= 0 )(x 2)(x+ 2)lnx= 0

Chapter 8. Integration techniques 8.1: Using integral tables. - UH Solution: The obvious choice here is u = x2 and dv = lnxdx. But this isnt so good because it will force us to integrate lnx. Rather let us write R x2 lnxdx = R (lnx)x2 dx, and let u = lnx,dv = x2 …

A Reduction Formula - MIT OpenCourseWare When using a reduction formula to solve an integration problem, we apply some rule to rewrite the integral in terms of another integral which is a little bit simpler. We may have to rewrite that …

Math Formulas: Integrals of Logarithmic Functions lnxn x dx = (lnxn)2 2n; (for n 6= 0) 11. Z lnx xm dx = lnx (m 1)xm 1 1 (m 1)2xm 1; (fot m 6= 1) 12. Z (lnx)n xm dx = (lnx)n (m 1)xm 1 + n m 1 Z (lnx)n 1 xm dx; (fot m 6= 1) 13. Z dx xlnx = lnjlnxj 14. …

Integral Test - dbenedetto.people.amherst.edu u= lnx du= 1 x dx and x= 2 )u= ln2 x= t)u= lnt The improper integral diverges. Finally, the Original Series also diverges by the Integral Test. 3. X1 n=1 e1 n n2 The related function f(x) = e1 x x2 …

Integration - practice questions Hint: use integration by parts with f = ln x and g0 = x4. Solution: If f = ln x, 0 1 then f = . Also if g0 = x4, then g = 1 x5. Hint: the denominator can be factorized, so you can try partial fractions, but …

Table of Integrals x2 lnx x2 4 (44) Z x2 lnxdx= 1 3 x3 lnx x3 9 (45) Z xn lnxdx= xn+1 lnx n+ 1 1 (n+ 1)2 ; n6= 1 (46) Z lnax x dx= 1 2 (lnax)2 (47) Z lnx x2 dx= 1 x lnx x (48) Z ln(ax+ b) dx= x+ b a ln(ax+ b) x;a6= 0 5 …

Chapter 9: Indeﬁnite Integrals - Chinese University of Hong Kong x2 + 2 3 x3 2 + x+ C Example 9.2.2. Find the function f(x) whose tangent line has slope 4x3 + 5 for each value of x and whose graph passes through the point (1, 10). Solution. The slope of …

Chapter 2. Methods of Integration - math.uwaterloo.ca To integrate a polynomial (or an algebraic) function times a logarithmic or inverse trigonometric function, try integrating by parts letting u be the logarithmic or inverse trigonometric function. …

Table of Integrals - UMD ©2005 BE Shapiro Page 3 This document may not be reproduced, posted or published without permission. The copyright holder makes no representation about the accuracy, correctness, or

Practice Integrals - University of Cambridge A facility with integration will help with the execution of more sophisticated problems in this and other courses in the Mathematics Tripos. This sheet is intended for self-study, and answers …

Integration by Parts - University of Notre Dame General Rule: When choosing u and dv, u should get \simpler" with di erentiation and you should be able to integrate dv. F(a).) 2 ln(x + 3)dx. 2 I To calculate R x dx, we use substitution with w …

Integrals of Exponential Functions. Integrals Producing Logarithmic ... We summarize the formulas for integration of functions in the table below. 1. Evaluate the following integrals. In problems (d), (k) and (n) a and b are arbitrary constants. 2. Recall that …

7.1 The Logarithm Deﬁned as an Integral Chapter 7. Integrals … lnx = Z x 1 1 t dt. Note. It follows from the deﬁnition that for x ≥ 1, lnx is the area under the curve y = 1/t for t ∈ [1,x]. All we can currently tell from the deﬁnition, is that lnx < 0 for x ∈ (0,1), ln1 = …

Integral X 2 Lnx

Integrating the Natural Logarithm: A Comprehensive Guide to ∫x²lnx dx

Links:

Converter Tool

Conversion Result:

Formatted Text:

Search Results: