What is Trig substitution: Definition and 117 Discussions

In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. In calculus, trigonometric substitution is a technique for evaluating integrals. Moreover, one may use the trigonometric identities to simplify certain integrals containing radical expressions. Like other methods of integration by substitution, when evaluating a definite integral, it may be simpler to completely deduce the antiderivative before applying the boundaries of integration.

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  1. L

    Trig Substitution and Volume of Solids

    2 Questions here! (I'm not exactly sure if it's allowed, but I want to avoid posting too many threads) Homework Statement 1)Find the following definite integrals by using a trigonometric substitution: d)1/2∫1dx/(√(2x-x2) Homework Equations x=asinu The Attempt at a Solution From...
  2. R

    Cal 2 integral / trig substitution

    Homework Statement I am asked to prove the following statement is correct integral (sqrt(a^2+x^2))/x dx = sqrt(a^2+x^2)-a log(a (sqrt(a^2+x^2)+a))+ C Homework Equations x = atanθ dx = (asecθ)^2 tan^2+1 = sec^2 The Attempt at a Solution got down to a (sec^2 θ a(√sec^2)dθ)/atanθ...
  3. P

    Trig substitution should be simple but it's driving me nuts

    Homework Statement \int x/sqrt{(x^{2}+4)} Homework Equations x=2tanx The Attempt at a Solution x=2tanx \int2tan\vartheta/\sqrt{tan^2\vartheta}+4 2/2 *\inttan/sec \intsin=-cos now is the part where i am stuck i know from using substitution that the answr should be...
  4. P

    Is Trig Substitution Necessary for Solving This Integral?

    first off, i can solve this problem easily with u-sub but the question asks to use trig sub. \int4x/(x2+1) then i have x=tan(theta) dx=sec2(theta) =\int4tan(theta)sec2(theta)/(tan2(theta)+1) =\int4tan(theta)d(theta) =-4ln(cos(theta))=-4ln(cos(arctan(x))) thanks for the help...
  5. W

    Applying Integration by Parts & Trig Substitution

    Homework Statement \int\sqrt{4+9x^{2}}dx Homework Equations Pythagorean Identities? The Attempt at a Solution I find it sort of cumbersome to use the special formatting here, so I hope it is okay that I just photocopied my work on paper. You can see how far I made it, but...
  6. B

    When integrating by trig substitution why do you use what you use?

    when integrating by trig substitution why do you use what you use?? for example int. (1+x^2)^0.5 dx why do you use x= tan u i mean obviously because it works, but if you didn't know it works how would you figure it out? i would think that you should use x=sinh u but I've been trying...
  7. H

    Integral with trig substitution

    Homework Statement Using trig substitution integrate x2/(x4+6x2+9) Homework Equations The Attempt at a Solution I'm almost completely positive I need to use partial fraction decomposition... doing this I end up having the integral of 1/(x2+3)-3/(x2+3)2 There aren't any square...
  8. H

    Integration using trig substitution

    Homework Statement Integrate (9x2-16)1/2/x4 Homework Equations The Attempt at a Solution I set 3x=4secy, 3dx=4secytany and (9x2-16)1/2=2tany. I then plugged that all into my integral and ended up with2tany*4secytany/(4/3secy)4 dy... (81/32) tan2y/sec3y dy... I solved for...
  9. C

    Integration: Trig Substitution

    My apologies. I'm not proficient with latex, and it is bogging my computer down for some reason today. Homework Statement intdx/\sqrt{(4-x^2)} [0, 2/sqrt{2} Homework Equations Trig Identity: a^2-a^2sin^2\theta The Attempt at a Solution In the interest of my own sanity I am going to...
  10. M

    Trig Substitution Homework: Int x^2 + 36/4x^2

    Homework Statement \int\frac{\sqrt{x^2 + 36}}{4x^2}}dx Homework Equations sqrt(a^2 + x^2) substitution for x = a tan theta The Attempt at a Solution I set x = 6 tan theta x^2 = 36 tan^2 theta dx = 6 sec^2 x \int\frac{\sqrt{36 + 36 tan \theta}}{144 tan \theta}dx \int\frac{\sqrt{36(tan...
  11. V

    Question about trig substitution intervals

    Homework Statement my professor tell me that when looking at the case ∫ √ (a^2 - x^2) , the trig substitution of course is asinϑ where -pi/2 ≤ ϑ ≤ pi/2. What I don't understand is why my professor tells me that when this term, √ (a^2 - x^2), is in the denomenator of the integrand that we must...
  12. Z

    Improper integrals and trig substitution

    Homework Statement It's been a couple of years since I've done real math, so I'm kinda stuck on this one. This is actually part of a physics problem, not a math problem - but I'm stuck on the calculus part. I'm trying to solve this guy: \int \limits_{-\infty}^{\infty}...
  13. K

    Calculating Sphere Volume Using Trig Substitution

    Homework Statement find the volume of a sphere of radius R by integrating over the primary volume elements in cartesian coordinates. Hint: use a trig substitution for your integral over dy Homework Equations The Attempt at a Solution I don't understand what the problem wants me...
  14. V

    Another trig substitution question

    Homework Statement sorry wait a few moments for the details, i hit post on accident prematurely ∫ √(1 + x^2)/x dx The Attempt at a Solution ∫ √(1 + x^2)/x dx x = tanϑ , dx sec^2ϑ dϑ -π/2 < ϑ < π/2 √(1 + x^2) = secϑ ∫ (secϑ * sec^2ϑ dϑ )/ tanϑ dϑ after using trig identities...
  15. V

    Question about where to start this trig substitution integral

    Homework Statement ∫ x^2√(a^2 - x^2) dx evaluate integral from 0 to a Homework Equations The Attempt at a Solution so i know the format of this problem requires the substitution of asinϑ -π/2 ≤ ϑ ≤ π/2 , but i don't know how to change the bounds of the integral...
  16. V

    Trig substitution integral(need a check on the solution)

    Homework Statement this is the first problem like this I've ever tried so take it easy!:redface: evaluate the integral I = ∫ x^3/√(16-x^2) dx from 0 to 2√3 The Attempt at a Solution - π/2 < ϑ < π/2 x = 4sinϑ , dx = 4cosϑdϑ *x = 2√3, x/4 = sinϑ , sinϑ = √3/2 , ϑ = π/3 x = 0 sinϑ = 0...
  17. B

    Inverse Trig Substitution Integral

    \int x^{2} \sqrt{4+x^{2}} dx I've already subbed in: x = 2tan\theta dx = 2sec^{2}\theta d\theta and I've gotten down to: 16 \int tan^{3}\theta sec^{3}\theta d\theta But now I have noo idea what to do! Can someone give me a hint?
  18. R

    Solve Indefinite Integral (x^3)sqrt((x^2)+4)dx

    Homework Statement Indefinite Integral (x^3)sqrt((x^2)+4)dx Homework Equations With an x= 2tan@ and dx= 2 (sec^2)@ d@ The Attempt at a Solution I get to 8(tan^3)@(sqrt((4tan^2)@+(8sec^2)@d@ Simplified down to 8(tan^3)@(sqrt((12tan^2)@+8d@ After that I'm stuck The...
  19. N

    Integration By Trig Substitution

    1. Homework Statement ∫√(4-x^2)/x dx 2. Homework Equations 3. The Attempt at a Solution a^2=4 u^2=x^2 ⇒ u=asinθ a=2 u=x x=2sinθ sinθ 2cosθ=√(4-x^2) dx=2cosθ dθ ∫√(4-4sin^2θ)/2sinθ 2cosθ dθ ∫2cos^2θ/2sinθ 2cosθ dθ 2∫cos^2θ/sinθ cosθ dθ Now what? I have been working...
  20. N

    Solving Trig Substitution Homework: ∫√(4-x^2)/x dx

    Homework Statement ∫√(4-x^2)/x dx Homework Equations The Attempt at a Solution a^2=4 u^2=x^2 ⇒ u=asinθ a=2 u=x x=2sinθ sinθ=x/2 (Our professor uses a triangle method which I won't draw) 2cosθ=√(4-x^2) dx=2cosθ dθ ∫√(4-x^2)/x dx=∫2cosθ/2sinθ dθ...
  21. C

    Integrals involving trig substitution

    Homework Statement I don't know how to solve these. How do you evaluate the integral of \frac{3dx}{\sqrt{3+X^2}}? I know you have to set x=atan\theta. our a is \sqrt{3} so x =\sqrt{3}tan\theta. That means dx=\sqrt{3}sec2\thetad\theta. I also made a right triangle using the information...
  22. D

    Integration by trig substitution

    Homework Statement the problem is INTEGRAL 6dz/(z^2(sqrt(z^2+9)) z^2+ a^2 , then z=a*tan@ where a here is 3, because 3^2=9 , i use @ here to represent theta substituting this for z; Int[6/(z^2(sqrt(z^2+9))]dz= Int[(6*3sec^2@d@)/(9tan^2@(sqrt(9tan^2@+9))]=...
  23. A

    What is the Appropriate Substitution for Solving the Integral of x/(x^2+2x+2)dx?

    integral of x/(x2+2x+2)dx first thing i did was complete the square to get x/((x+1)2+1 i tried then having x+1 = tanx but that didnt work out because of the x on top i can't just set w = x+1 what would the right substitution be? any hints or help would be appreciated
  24. James889

    Inverse trig substitution integral

    Hi, I need to integrate the following: \int \frac{x^2}{\sqrt{9-x^2}} So let x = 3sin\theta \frac{dx}{d\theta} = 3cos\theta So i now have the integral of \frac{9sin^2\theta \cdot 3cos\theta}{3cos\theta} How do i go about the integration from here? parts?
  25. B

    Solving Integrals with Trig Substitution - 1/(25-x^2)

    Hi Everyone! I just need some guidance on this problem. I seem to have trouble what integration technique I need to use on integrals of this type. Homework Statement integrate 1/(25-x^2) Homework Equations sqrt(a^2-u^2) arcsin(u/a) The Attempt at a Solution Would I be...
  26. B

    Trig Substitution: Solving Integrals with sec^3Θ

    http://img708.imageshack.us/img708/8897/symimage.gif so I did x=atanΘ. which is x=3tanΘ and dx is 3sec^2\Theta. Then it is \sqrt[]{9tan^2\Theta+9}*3sec^2\Theta which evaluates after factoring to \sqrt[]{9sec^2\Theta}*3sec^2\Theta which is then 3sec\Theta*3sec^2\Theta If i take the 9...
  27. W

    Integration With Trig Substitution Calc II

    Homework Statement \int \frac{\sqrt{196 x^2-144}}{x} dx Homework Equations The Attempt at a Solution I first rewrote the integral... \int \frac{\sqrt{(14x)^2-12^2}}{x} dx Then I let... 14x=12sec\theta thus... x=6/7sec\theta dx=6/7sec \theta tan \theta d \theta My...
  28. R

    Trig Substitution (?) Integral

    Homework Statement The answer is: The Attempt at a Solution I tried trig substitution, letting x =\sqrt{2}tan(\theta) and using the identity 1+tan^{2}=sec^{2}(\theta), but couldn't get to the answer. Thanks for the help.
  29. M

    How can I evaluate this integral using trig substitution?

    Homework Statement evaluate the integral using the indicated trig substitution. sketch the corresponding right triangle. integral of(1/(x^2 sqrt(x^2 - 9)) Homework Equations integral of(1/(x^2 sqrt(x^2 - 9)) The Attempt at a Solution at first glance this seemed really easy...
  30. S

    Solving Int. with Trig Substitution for Beginners

    Homework Statement \int\frac{\sqrt{1+x}+\sqrt{1-x}} { \sqrt{1+x}-\sqrt{1-x}}{dx} Homework Equations I believe trig substitution can be used here. I'm not very good at calculus only beginning to take calc classes, and guideance would be wonderful. because i want to get better. The...
  31. S

    Trig Substitution for Integrating (y-x)/(a^2(y)-a^2(x))^b - Homework Help

    Homework Statement I want to integrate: \int^y_0\frac{y-x}{(a^2(y)-a^2(x))^b}dx Homework Equations a2(y) means that a is a function of y. similarly for [a(x)]2. so [a(x)]2 is a functions that depends on x. The Attempt at a Solution I tried integration by parts: let u=y-x so...
  32. Q

    What Are the Correct u and du Values for These Trig Substitution Problems?

    I really needed some help with these trig substitution problems. Please write the u and du values for each. I'd really really appreciate it ! 1) Integrate [e^x/(e^x +1)] dx 2) Integrate [1/(e^2x +1)] dx 3) Integrate [(1+x)/(1+x^2)] dx Thank you Again !
  33. M

    Integration by Trig Substitution

    I've attempted solutions at this in two different manners and found myself stuck both ways. I'll show you the way that seems to make progress. The other way involves not factoring the junk under the radical in the denominator. Homework Statement \int^{1}_{0} \frac{3x^2 -1}{\sqrt{x-x^3}} dx...
  34. D

    Solve Trig Substitution Homework with P

    Homework Statement If sin54=P, express the following in terms of p, without using a calculator. 1] cos18 2] \frac{tan27+cot63}{1+tan207.cot117} Homework Equations The Attempt at a Solution 1] cos18=sin(90-18) =sin(72) sin72=sin(180-108) =sin(180-108)...
  35. B

    Difficult integral for Trig Substitution

    ok i have been studying the in-depth processes of trigonometric substitution with integrals and this problem has me frusterated. \int x^2\sqrt{(x^2-4)} dx The evaluation is clear (from an old Table of Integrals I found), but the derivation is not at all clear, which is what i want to know. I...
  36. K

    Integration using Hyperbolic Trig substitution

    Homework Statement Evaluate: \int\\{1}/{\sqrt{x^2-1}} dx between -3, -2 I know I'm supposed to use hyperbolic substitution in the question. Homework Equations edit: cosh^2(t) - sinh^2(t) = 1 The Attempt at a Solution let x = -cosht, inside the integral let dx = sinh(t) dt int (...
  37. mvantuyl

    Integration by trig substitution

    Homework Statement Integrate: \int\sqrt{1 - 9t^{2}}dt Homework Equations The Attempt at a Solution t = 1/3 sin x dt/dx = 1/3 cos x dt = 1/3 cos x dx 3t = sin x 1/3 \int\sqrt{1 - sin^{2} x} cos x dx 1/3 \int cos ^{2}x dx 1/3 \int(1 + cos 2x) / 2 dx 1/6 \int1 + cos 2x dx...
  38. mvantuyl

    Integration by trig substitution

    Homework Statement Integrate: \int\sqrt{1-9t^{2}}dt Homework Equations The Attempt at a Solution t = 1/3 sin\Theta dt/d\Theta = 1/3 cos\Theta dt = 1/3 cos\Thetad\Theta 3t = sin\Theta 1/3\int\sqrt{1-sin^{2}}\Theta cos\Thetad\Theta 1/3\int cos^{2}\Thetad\Theta 1/3\int(1 +...
  39. J

    Can someone please check my answer: Trig Substitution problem

    I tried to use the latex reference but it was way too complicated for me. Can you see the attachment. Thank you!
  40. D

    Trig Substitution Homework: Solving \int{cos^4 6x sin^3 6x dx}

    Homework Statement \int{cos^4 6x sin^3 6x dx} Homework Equations The Attempt at a Solution I've gotten this far but now I'm stuck: \int{cos^4 6x sin 6x sin^2 6x dx} = \int{cos^4 6x}*(\frac{1-cos 12x}{2})sin 6x dx
  41. P

    Integral, Trig Substitution - Stuck

    Homework Statement Use trigonometric substitution to evaluate \int{\frac{x^2}{\sqrt{9-x^2}}}dx The Attempt at a Solution Let x=3\sin\theta then dx=3\cos\theta d\theta \int{\frac{x^2}{\sqrt{9-x^2}}}dx =\int{\frac{9\sin^2\theta}{3\sqrt{1-\sin^2\theta}}}\ 3\cos\theta \ d\theta...
  42. K

    Trig substitution integration problem, test in 1hr 30min

    Homework Statement √(9-x²) / (x²) Homework Equations Just trig substitution The Attempt at a Solution Ok, for trig sub I did u=asinΘ x=3sinΘ 9-x²=9-9sin²=9(1-sin²Θ) so putting it into the equation √9cos²Θ=3cosΘ/x^2 where do I go from here, I tried getting help at Math...
  43. N

    Integrate using trig substitution

    Homework Statement int sqrt 8x-x^2 Homework Equations trig sub The Attempt at a Solution complete the square integral becomes int sqrt 16-(x-4)^2 let x-4= 16sin(Q) sqrt 16-(x-4)^2 =sqrt 16-256sin^2(Q) dx= 16cos(Q)dQ =...
  44. P

    Integral with trig substitution

    [SOLVED] Integral with trig substitution Homework Statement Find \int(x^3)/\sqrt{x^2-9} Homework Equations Trig substitution. sin^2 + cos^2 =1, and other things that you can figure out from that. Half angle formula, cos^2\theta=(1+cos(2\theta) )*.5 The Attempt at a Solution...
  45. O

    Integration with Trig Substitution

    Homework Statement Ok, so I was doing a problem on the electric field strength of a continuous charge distribution and I arrived at this seemingly easy integral \int \frac{1}{({l^2+a^2})^\frac{3}{2}} dl sorry the latex is lagging badly, you can see the correct integral by clicking on it. it...
  46. B

    Integration using Trig Substitution

    Homework Statement \int \frac{cosx dx}{\sqrt{1 + sin^{2}x}} Homework Equations Expression: \sqrt{a^{2} + x^{2}} Substitution: x = a*tan\Theta Identity: 1 + tan^{2}\Theta = sec^{2}\Theta The Attempt at a Solution I have tried using Trig Substitution, but I end up getting an equation much...
  47. N

    Trig Substitution for Integrating sin^3(x)/cos(x) - Homework Help

    Homework Statement \int ((sin(x))^3/(cos(x)) )*dx The Attempt at a Solution alright i have been trying to use u= cosx -du = sinx but it doesn't make sense bause there is still a sinx^2 to account for so i know i need to make a trig substitution but i can't figure out the appropriate...
  48. Q

    Trig Substitution: Solving for the Missing Identity

    [SOLVED] More trig substitution help... I've looked at this problem about 3 times and still can't figure it out...where identity did they use to substitute out the part in the red box? Thanks for the help
  49. F

    Where Did I Go Wrong with This Trig Substitution for x^3/sqrt(1-x^2)?

    For the integral \int frac{x^3}{sqrt{1-x^2}} dx} ==> okay... what I meant was: int of x^3 over sqrt(1-x^2) --I trig substitute to get sin^(3)(x)cosxdx over cos x and end up with sin^3(x)...this is obviously wrong, can anyone point out what i did wrong?
  50. tony873004

    Solve Trig Substitution Integral: \int_{}^{} {\frac{x}{{\sqrt {3 - x^4 } }}dx}

    [SOLVED] trig substitution This is from the class notes. Evaluate the integral: \int_{}^{} {\frac{x}{{\sqrt {3 - x^4 } }}dx} \begin{array}{l} u = x^2 ,\,\,du = 2x\,dx\,\, \Leftrightarrow \,\,dx = \frac{{du}}{{2x}} \\ \\ \int_{}^{} {\frac{{x^1 }}{{\sqrt {3 - x^4 } }}dx} =...
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