Calc II integration by trig sub

  • Thread starter Thread starter Rock32
  • Start date Start date
  • Tags Tags
    Integration Trig
Click For Summary
SUMMARY

The integral from \(\sqrt{3}\) to \(3\) of \(\frac{\sqrt{x^2 + 9}}{x}dx\) was evaluated using the substitution \(x = 3\tan(u)\). The user derived a solution involving secant functions, specifically \(\sec(\pi/4) - \sec(\pi/6)\), but found discrepancies with the textbook answer, which is \(\frac{1}{6}\ln((3-2\sqrt{2})(7+4\sqrt{2}))\). The discussion highlights the importance of verifying each step in the substitution process to ensure accuracy in integral evaluation.

PREREQUISITES
  • Understanding of integral calculus, specifically techniques of integration.
  • Familiarity with trigonometric substitutions in integrals.
  • Knowledge of secant and inverse trigonometric functions.
  • Ability to manipulate logarithmic expressions and evaluate limits.
NEXT STEPS
  • Review trigonometric substitution techniques in integral calculus.
  • Learn about the properties and applications of secant functions.
  • Study the process of verifying integral solutions step-by-step.
  • Explore logarithmic identities and their use in calculus problems.
USEFUL FOR

Students studying calculus, particularly those focusing on integration techniques, as well as educators looking for examples of trigonometric substitution in practice.

Rock32
Messages
12
Reaction score
0

Homework Statement



Find the integral from 31/2 to 3 of 1/(x) ((x2 +9))1/2


I tried to substitute x= 3tanu, and I get down eventually to a solution that is (sec(pi/4) - sec(pi/6))

I realize integrals can have many forms, but the form in the back of my book isn't even close to sec(arctan(3/3)-sec(arctan(31/2/3). My answer doesn't match the books. Maybe its a domain problem?

The books answer is: 1/6ln((3-2*21/2)(7+4*31/2

Sorry, I'm exhausted, and I do not have the energy to type out how I got from the tangent substitution to my final value.

Any help would be great.
 
Physics news on Phys.org
Is this your integral?
\int_{\sqrt{3}}^3 \frac{\sqrt{x^2 + 9}}{x}dx
Your substitution looks to be the correct one, but if you're not getting the same answer as in your text, possibly you made a mistake after doing the substitution. It's also possible, but less probable, that the book's answer is wrong.
 

Similar threads

Integrate [cosec(30°+x)-cosec(60°+x)] dx in terms of tan x
  • · Replies 3 ·
Replies
3
Views
2K
"Gabriel's Horn" - A 3-D cone formed by rotating a curve
  • · Replies 3 ·
Replies
3
Views
2K
Fourier Transform - Solutions Error?
  • · Replies 5 ·
Replies
5
Views
2K
Trouble seeing the difference between autograder answer and my own
  • · Replies 2 ·
Replies
2
Views
2K
Calculate this Integral around the Circular Path using Green's Theorem
  • · Replies 3 ·
Replies
3
Views
2K
Integration problem ∫1/(√3 sinx+ cosx) dx
  • · Replies 8 ·
Replies
8
Views
2K
The integral of (sin x + arctan x)/x^2 diverges over (0,∞)
  • · Replies 5 ·
Replies
5
Views
3K
Double Integral of function in region bounded by two circles
  • · Replies 19 ·
Replies
19
Views
3K
Calculating radius of gyration of plane figure about x-axis
  • · Replies 5 ·
Replies
5
Views
3K
U-Substitution in trig integral
  • · Replies 3 ·
Replies
3
Views
3K