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. deuteron

    Potential of a rotationally symmetric charge distribution

    First, we rewrite the term ##|\vec r-\vec r_q|## in the following way: $$|\vec r-\vec r_q|= \sqrt{(\vec r-\vec r_q)^2} = \sqrt{\vec r^2 + \vec r_q^2 -2\vec r\cdot\vec r_q} = \sqrt{r^2 + r_q^2 -2rr_q\cos\theta}$$ Due to rotational symmetry, we go to spherical coordinates: $$\phi_{e;\vec r_q} =...
  2. Zack K

    Using Trig Substitution in Trig Integration

    Homework Statement Integrate: $$\int \frac{dx}{x^2\sqrt{4-x^2}}dx$$ Homework EquationsThe Attempt at a Solution I got to the final solution ##\int \frac{dx}{x^2\sqrt{4-x^2}}dx=-\frac{1}{4}cot(arcsin(\frac{1}{2}x))##. But It's the method where you transform that to the solution...
  3. karush

    MHB 7.3.5 Integral with trig substitution

    $\textsf{Evaluate the integral}$ $$I=\displaystyle\int\frac{x^2}{\sqrt{9-x^2}}$$ $\textit{from the common Integrals Table we have}$ $$\displaystyle I=\int\frac{u^2}{\sqrt{u^2-a^2}} \, du =\frac{u}{2}\sqrt{u^2-a^2}+\frac{a^2}{2} \ln\left|u+\sqrt{u^2-a^2}\right|+C$$...
  4. A

    Trigonometric substitution, What am i doing wrong?

    Homework Statement Homework Equations The Attempt at a Solution Here is my answer, i get 1/24 For my first step i divided both terms under the radical by 4, then split 1/4 into (1/2)2, i saw something very similar in my book so i did the same thing, but i just realized this has to be...
  5. Draconifors

    Triple integral using cylindrical coordinates

    Homework Statement The first part of the question was to describe E the region within the sphere ##x^2 + y^2 + z^2 = 16## and above the paraboloid ##z=\frac{1}{6} (x^2+y^2)## using the three different coordinate systems. For cartesian, I found ##4* \int_{0}^{\sqrt{12}} \int_{0}^{12-x^2}...
  6. uchuu-man chi

    I Need a little push on this integral using trig substitution.

    ∫x2√(3+2x-x2) dx Here's what I've already done: completed the square ∫x2√(4-(x-1)2) dx (x-1) = 2sinθ sinθ = (x-1)/2 x = 2sinθ+1 dx = 2cosθ dθ trig sub + pulled out constants 4∫(2sinθ+1)2√(1-sin2θ)cosθ dθ trig identity 4∫(2sinθ+1)2√(cos2θ)cosθ dθ 4∫(2sinθ+1)2(cos2θ)dθ expanded + trig...
  7. J

    Why Does Trig Substitution Yield Different Integral Results?

    Homework Statement ∫8cos^3(2θ)sin(2θ)dθ Homework EquationsThe Attempt at a Solution rewrote the integral as: 8∫(1-sin^2(2θ))sin(2θ)cos(2θ)dθ u substitution with u=sin(2θ) du=2cos(2θ)dθ 4∫(1-u^2)u du= 4∫u-u^3 du 4(u^2/2-u^4/4)+C undo substitution and simplify 2sin^2(2θ)-sin^4(2θ)+C The book...
  8. T

    MHB Trig substitution question

    $$\int_{}^{} \frac{1}{x\sqrt{x^2 + 16}} \,dx$$ I can set $x = 4 tan\theta$. Thus $dx = 4 sec^2 \theta d\theta$ So, plug this into the first equation: $$\int_{}^{} \frac{4 sec^2 \theta }{4 tan\theta \sqrt{16 tan^2\theta + 16}} \,d\theta$$ Then, $$\int_{}^{} \frac{ sec^2 \theta }{4 tan\theta...
  9. T

    MHB Integral using trig substitution

    I have $$\int_{}^{} \frac{1}{\sqrt{1 - x^2}} \,dx$$ I can let $x = \sin\left({\theta}\right)$ then $dx = cos(\theta) d\theta$ then: $$\int_{}^{} \frac{cos(\theta) d\theta}{\sqrt{1 - (\sin\left({\theta}\right))^2}}$$ Using the trig identity $1 - sin^2\theta = cos^2\theta$, I can simplify...
  10. B

    Trig Substitution Problem w/ tan substitution

    Homework Statement Under #3 Homework Equations Trig identities The Attempt at a Solution The picture attached is my attempt. The square in the upper upper left is the problem and the one in the lower right is my solution. I'm seeing that I'm getting the wrong answer, but not how.
  11. Battlemage!

    Integral of dn/(n^2 - 4) using trig substitution with sine

    Homework Statement \int_{}^{∞} \frac{1}{n^2 - 4} dn Homework Equations I'm trying to do this a way that it isn't usually done. Normally this is done with partial fractions. I'm trying to do it by using trig substitution using sine, which requires some algebraic manipulation. For some reason...
  12. karush

    MHB How Does Trigonometric Substitution Simplify the Integral Calculation? $a=0 \ \ b=12$ $\displaystyle I_{49}=\int_{a}^{b} \frac{dx}{\sqrt[]{144-x^2}} \, dx = \arcsin{\left[\frac{1}{12}\right]} \\$ $ \text{use identity} $ $\sin^2\theta+\cos^2\theta = 1 \Rightarrow 1-\cos^2\theta=\sin^2\theta \\$ $\text{x substituion} $ $\displaystyle...
  13. T

    MHB Integral using trig substitution

    I have this integral: $$\int_{}^{}\frac{1}{x^2 - 9} \,dx$$ I believe I can use trig substitution with this so I can set $x = 3 sec\theta$ Evaluating this, I get $$ln|\csc\left({\theta}\right) - \cot\left({\theta}\right)| + C$$ Since $x^2 - 9 = 9sec^2\theta - 9$, then $\frac{x^2 - 9}{3} =...
  14. karush

    MHB Can Trig Substitution be used to Solve Trigonometric Integrals?

    {8.7.4 whit} nmh{962} $$\displaystyle I=\int \sin\left({t}\right) \cos\left({2t}\right) \ dt $$ substitution $u=\cos\left({t}\right) \ \ \ du=-\sin\left({t}\right) \ dt \ \ \ \cos\left({2t}\right) =2\cos^2 \left({t}\right)-1 $ $\displaystyle I=\int \left(1-2u^2 \right) du \implies...
  15. karush

    MHB Trig Substitution for Whitman 8.4.8: Complete the Square

    Whitman 8.4.8 Trig substitution? Whitman 8.4.8 Complete the square.. \begin{align*} \int\sqrt{x^{2}-2x}dx &=\int\sqrt{x^{2}-2x+1-1}dx\\ &=\int\sqrt{(x-1)^{2}-1^{2}}dx\\ &=\int\sqrt{U^{2}-1^{2}}dx\\ \end{align*} Was wondering what substation best to use...
  16. RoboNerd

    I Question on basic trig substitution with x = sin theta

    Say I have the integral of [ 1 / ( sqrt( 1 - x^2) ] * dx . Now I was told by many people in videos that I substitute x = sin theta, and this has me confused. Wouldn't I need to substitute x = cos theta instead? as x = cos theta on the unit circle instead of sin theta? Thanks in advance for...
  17. N

    Integrate √1+x^2 - Solutions & Explanations

    1. The problem is as follows: ∫(√1+x^2)dx/(x) 2. Using trig sub --> x = atanΘ with a = √1 = 1. So x = tanΘ and dx = sec^2ΘdΘ. 3. Picture included of attempted solution. I tried u substitution with both u = secΘ and u=tanΘ but didn't have the right du...
  18. Ethan Godden

    Integration with trig substitution

    Homework Statement The problem is the integral attached Homework Equations sec2(u)=(1+tan2(x)) a2+b2=c2 ∫cos(u)=-sin(u)+C The Attempt at a Solution The solution is attached. I am wondering if someone could give me a hint where I went drastically wrong or where I possibly dropped a negative...
  19. UMath1

    Can Trig Substitution with Cosine be Used Instead of Sine?

    I was wondering if you could do a trig substitution with cosine instead of sine. All the textbooks I have referred to use a sine substitution and leave no mention as to why cosine substitution was not used. It seemed that it should work just the same, until I tried it for the following Fint...
  20. R

    Can you skip trig substitution?

    Hi, I'm currently taking ap calc bc as a senior in high school. Since trig sub and power reduction formula is not apart of the ap curriculum our class is skipping it. Assuming I pass the test and get credit for it, I will probably skip calc 2 in college. If I continue to study math and physics...
  21. T

    Integral Trig Substitution Question

    I just have a few questions. When using a trig substitution does it have to be under a radical ? eg, suppose I wanted to integrate (x2)/(x2-9), I used a trig substitution of x = 3sec(t) and got the wrong answer and so apparently I had to use partial fractions
  22. Dethrone

    MHB When to use a hyperbolic trig substitution in integration problems?

    I read somewhere that: sqrt(a^2-x^2), you can use x = asinx, acosx sqrt(a^2+x^2), you can use x = atanx (or acotx), asinhx sqrt(x^2-a^2), you can use x = asecx (or a cscx), acoshx When would it be beneficial to use a hyperbolic trig substitution as oppose to the regular trig substitutions (sin...
  23. S

    MHB Solving an Integral Using Trig Substitution

    \int \frac{81/100}{100x^2 + 81} so I know this is 1 + tan^2\theta but how do I implement it? isn't it a^2 + x so should i let x = 100tan\theta?
  24. J

    Using trig substitution or partial fractions?

    When would you use trig substitution vs. partial fractions? I know partial fractions is when you have a polynomial over a polynomial, but some of the problems in the trig substitution section in my book had polynomial over polynomial and used trig substitution?
  25. E

    Trig substitution integration?

    Homework Statement Integrate dx/((x^2+1)^2) Homework Equations Tan^2=sec^2-1 The Attempt at a Solution So I let x=tanx then dx=sec^2x Then plugging everything in; Sec^2(x)/(tan^2+1)^2 So it's sec^2/(sec^2x)^2) which is sec^2x/sec^4x Canceling out the sec^2 gives...
  26. S

    Help with Trig Substitution Integral Problem

    Homework Statement Question is attached in this post.Homework Equations Question is attached in this post. The Attempt at a Solution I've solved the problem via using x=asinθ where a=1 I've been able to integrate the problem to the point where I get cos^2(θ)/sin^2(θ), but can't seem to...
  27. T

    Trig substitution into integrals

    I was testing for convergence of a series: ∑\frac{1}{n^2 -1} from n=3 to infinity I used the integral test, substituting n as 2sin(u) so here's the question: when using the trig substitution, I realized the upperbound, infinity, would fit inside the sine. Is it still possible to make...
  28. A

    MHB How Can I Integrate x*sqrt(1-x^4) Using Trig Substitution?

    how do i go about integrating x*sqrt(1-x^4)?? i have no idea
  29. A

    MHB Integrate sqrt(1 + x^2) / x: Trig Substitution

    how do u integrate sqrt(1 + x^2) / x?? i reduced this to sec^3(u)/tan(u) but how do i go from here??
  30. G

    Trig Substitution for ∫x^3/(4x^2+9)^(3/2): Solving the Insanity

    ∫x3 ---------------------- (4x2 + 9) 3/2 According to my book this is a trig substitution integral. The normal procedure is to substitute atanθ for x when one has a square root w an argument of the form x^2 + a^2. Because the argument of the square root is 4x2 + 9, as opposed to simply x2...
  31. T

    Trig Substitution: Solving Homework Equations

    Homework Statement Homework Equations The Attempt at a Solution This isn't really a traditional question, but can someone explain to me how substituting u = tan^-1(x/y) got to that final value? I'm trying to understand this for an exam coming up.
  32. Feodalherren

    Trig Sub Homework: Did I Make a Mistake?

    Homework Statement Did I make a mistake here somewhere? The solution in the back of the book is completely different. Seems like they used trig sub one step later or something. I can't find any error in my logic. Test coming up soon and I'm confused and panicking -_-! Actually I just found a...
  33. F

    Theta ranges for trig substitution

    My professor, when doing trig substitution in lecture, always defines theta between certain intervals and when he takes the square root, he adds an absolute value bar to the trig function and then makes sure its positive through the interval. For practical purposes, is it necessary to go through...
  34. W

    Area & Centroid of Region Bounded by arcsinx & x=1/2

    Homework Statement consider the region bounded by the graphs of y=arcsinx, y=0, x = 1/2. a) find the area of the region. b) find the centroid of the region.Homework Equations \displaystyle\int_0^{1/2} {arcsinx dx} u=arcsinx; du = \frac{1}{1-x^2}dxdv=dx ; v=x xarcsinx]^{1/2}_{0} -...
  35. H

    Integral with trig substitution

    Homework Statement ∫(x+1)/((x^2+1)^2) Homework Equations The Attempt at a Solution I have been able to separate this into 2 ∫x/(x^2+1)^2 dx which i found to be equal to (1/2)arctanx and ∫1/(x^2+1)^2 dx which i am unable to find What i did was sub in x=tanθ and dx=sec^2(θ)dθ, and with...
  36. W

    Evaluate Trig Subs Integral w/ e^x = sin∅

    Homework Statement Use a trigonometric substitution to evaluate the integral. Homework Equations \int e^x\,dx /\sqrt{1-e^2x} The Attempt at a Solution e^x = sin∅ x=lnsin∅ dx=cos∅/sin∅ \frac{sin∅*cos∅}{sin∅*\sqrt{1-(sin∅)^2}} \int sin∅cos∅ / sin∅(cos∅)\,d∅...
  37. W

    Integrate x^3/(x^2 - 16) with Trig Substitution

    Homework Statement evaluate the integral. Homework Equations integral (x^3 / (x^2 - 16) The Attempt at a Solution x=4sec∅ dx=4sec∅tan∅d∅ 1. i substituted those values in, and then split sec^4∅ into sec^2∅ and (1+tan^2∅). 2. integral 16 (1/u) du + integral 16 (u) du. 3. end...
  38. K

    How to Solve Integrals Using Trig Substitution?

    Homework Statement integral of dx/((9-(x^2))^(3/2)) A = 0, B = 3/2 Homework Equations Trigonometry Substitutions 3. The Attempt at a Solution : I am stuck with this question. So far, I got (1/9)integral of (1/cos^2(θ)) dθ
  39. I

    Trig Substitution for Integrating \frac{dx}{\sqrt{x^{2}+16}}

    Homework Statement \int \frac{dx}{\sqrt{x^{2}+16}}Homework Equations The Attempt at a Solution x=4tan\theta dx=4sec^{2}\theta d\theta Therefore: \int \frac{4sec^{2}\theta d\theta}{\sqrt{16tan^{2}\theta +16}} = \int \frac{sec^{2}\theta d\theta}{\sqrt{tan^{2}\theta+1}} \int \frac{sec^{2}\theta...
  40. M

    Nasty Integral - Help with Trig Substitution

    Homework Statement \int_{-p}^{p} \frac{2p}{(1+v^2)\sqrt{p^2 + v^2 +1 }} dv Homework Equations 1 + \tan{\theta}^2 = \sec{\theta}^2 The Attempt at a Solution I thought the best way to go about this was to rename some constants. Let \alpha^2 = 1 + p^2 so that we are left with...
  41. B

    Trig substitution step (I think)

    Homework Statement I'm stuck at an attempt to solve an integration step. I think I'm supposed to trig substitute? Homework Equations http://img685.imageshack.us/img685/9158/unavngivetn.png It is the second to third equation I'm having a hard time with The Attempt at a Solution From second...
  42. B

    Is Trig Substitution Needed for This Integral?

    Homework Statement \int\frac{1}{\sqrt{16-x^2}}dx Homework Equations csc\theta=\frac{4}{\sqrt{16-x^2}} 4cos\theta=x -4sin\theta d\theta=dx \theta=arccos(\frac{x}{4}) The Attempt at a Solution Using these facts, I concluded that the integral, after all of the substitution, was...
  43. B

    Trig substitution ∫(4x^3)/√(x^2+4)

    Homework Statement ∫(4x^3)/√(x^2+4)dx Homework Equations The Attempt at a Solution So, I let x= 2tanθ dx= 2sec^2θ dθ So, √(4tan^2(θ)+4)=2secθ ∫(4x^3)/√(x^2+4)dx=∫((32tan^3(θ))/(2secθ))2sec^2(θ)dθ. Would it go to ∫16tan^3(θ)2sec(θ)dθ or ∫32tan^3(θ)sec(θ)dθ
  44. D

    Integral by Trig Substitution, Calc 2

    Homework Statement The definite integral of ∫(x^2 √(a^2-x^2) dx from 0 to a Homework Equations The Attempt at a Solution So i don't need actual help with this problem. I got the answer, (π*a^4)/16 and I verified with the back of the book. The question I have is whether this...
  45. O

    Small trig substitution problem.

    Homework Statement I was working on a problem set involving greens theorem and I came across this peculiar trig substitution. I was just wondering how it came about as I couldn't find anything like it on Wikipedia's page. sin^4(t)cos^2(t) + cos^4(t) sin^2(t) = cos^2(t)sin^2(t) The...
  46. F

    Integral involving trig substitution

    Hello, I am trying to integrate 1/(x^2-1). Apparently this can be solved by using trig substitution involving tan ? Can someone please help me to understand how to go about it. Thanks kindly for any help.
  47. B

    Tricky Integration with Trig Substitution

    Homework Statement Evaluate. \int(4-y)\sqrt{4-y^{2}}dy I have the solution using CAS software here: 2y\sqrt{4-y^{2}}+8sin^{-1}\frac{y}{2}+\frac{1}{3}(4-y^{2})^{3/2} but I need to do this by hand. I have researched the usual trig methods but am having some difficulty. Can...
  48. S

    Trig substitution: integrate sqrt(16+x^2) over x

    Homework Statement \int\frac{\sqrt{16+x^{2}}}{x} Homework Equations The Attempt at a Solution set x=4tant dx=4sec^{2}t dt so after plugging in and using a quick trig identity I get: \int\frac{16(sec^{2}t)*4sec^{2}t dt}{4tant} Then after a quick cleanup: 16*\int...
  49. ArcanaNoir

    Trig substitution integral (I hope)

    Homework Statement I got to a place in a problem where I need to do a sticky integral, and I'm hoping I can use a trig substitution. If not, I will need to solve the main problem another way :( \int_0^\infty \sqrt{1+(e^{-\theta })^2} \; \mathrm{d} \theta Homework Equations 1+\tan ^2...