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.
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:
$$\phi_{e;\vec r_q} =...
Homework Statement
Integrate: $$\int \frac{dx}{x^2\sqrt{4-x^2}}dx$$
Homework EquationsThe Attempt at a Solution
I got to the final solution ##\int \frac{dx}{x^2\sqrt{4-x^2}}dx=-\frac{1}{4}cot(arcsin(\frac{1}{2}x))##. But It's the method where you transform that to the solution...
$\textsf{Evaluate the integral}$
$$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$$...
Homework Statement
Homework Equations
The Attempt at a Solution
Here is my answer, i get 1/24
For my first step i divided both terms under the radical by 4, then split 1/4 into (1/2)2, i saw something very similar in my book so i did the same thing, but i just realized this has to be...
Homework Statement
The first part of the question was to describe E the region within the sphere ##x^2 + y^2 + z^2 = 16## and above the paraboloid ##z=\frac{1}{6} (x^2+y^2)## using the three different coordinate systems.
For cartesian, I found ##4* \int_{0}^{\sqrt{12}} \int_{0}^{12-x^2}...
Homework Statement
∫8cos^3(2θ)sin(2θ)dθ
Homework EquationsThe Attempt at a Solution
rewrote the integral as:
8∫(1-sin^2(2θ))sin(2θ)cos(2θ)dθ
u substitution with u=sin(2θ) du=2cos(2θ)dθ
4∫(1-u^2)u du= 4∫u-u^3 du
4(u^2/2-u^4/4)+C
undo substitution and simplify
2sin^2(2θ)-sin^4(2θ)+C
The book...
I have
$$\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}}$$
Using the trig identity $1 - sin^2\theta = cos^2\theta$, I can simplify...
Homework Statement
Under #3
Homework Equations
Trig identities
The Attempt at a Solution
The picture attached is my attempt. The square in the upper upper left is the problem and the one in the lower right is my solution. I'm seeing that I'm getting the wrong answer, but not how.
Homework Statement
\int_{}^{∞} \frac{1}{n^2 - 4} dn
Homework Equations
I'm trying to do this a way that it isn't usually done. Normally this is done with partial fractions. I'm trying to do it by using trig substitution using sine, which requires some algebraic manipulation. For some reason...
I have this integral:
$$\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} =...
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...
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...
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...
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...
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...
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...
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
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...
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?
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...
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...
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...
∫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...
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.
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...
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...
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} -...
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...
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∅...
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...
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
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...
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...
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...
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
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...
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...
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.
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...
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...
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...