just how you combine \$frac1sqrt1+x^2\$ using adhering to substitution? \$1+x^2=t\$ \$Rightarrow\$ \$x=sqrtt-1 Rightarrow dx = fracdt2sqrtt-1dt\$... Now I"m stuck. I don"t know exactly how to proceed using substitution rule.

By the substitution you said you get\$\$int frac12sqrtt(t-1) ,dt=int frac1sqrt4t^2-4t ,dt=int frac1sqrt(2t-1)^2-1 ,dt\$\$Now the substitution \$u=2t-1\$ seems reasonable.

You are watching: Integral 1/sqrt(1-x^2)

However your initial integral can likewise be addressed by\$x=sinh t\$ and also \$dx=cosh t, dt\$ which gives\$\$int fraccosh tcosh t , dt = int 1, dt=t=operatornamearcsinh x = ln (x+sqrtx^2+1)+C,\$\$since \$sqrt1+x^2=sqrt1+sinh^2 t=cosh t\$.

See hyperbolic functions and also their inverses.

If friend are familiar (=used to manipulate) through the hyperbolic features then \$x=asinh t\$ is worth trying anytime you check out the expression \$sqrta^2+x^2\$ in her integral (\$a\$ gift an arbitrarily constant).

re-publishing
mention
monitor
edited Sep 23 in ~ 7:00

cutting board Andrews
answer Aug 5 "12 at 14:00

boy name SleziakMartin Sleziak
\$endgroup\$
6
\$egingroup\$ just how do you acquire from \$int frac1sqrt1+x^2 dx\$ to \$int frac1cosh tdx=int fraccosh tcosh tdt\$? \$endgroup\$
–user2723
Aug 5 "12 at 14:27

| present 1 more comment
13
\$egingroup\$
A variant of the hyperbolic function substitution is to let \$x=frac12left(t-frac1t ight)\$. Note that \$1+x^2=frac14left(t^2+2+frac1t^2 ight)\$.

So \$sqrt1+x^2=frac12left(t+frac1t ight)\$. That was the whole allude of the substitution, it is a rationalizing substitution that renders the square root simple. Also, \$dx=frac12left(1+frac1t^2 ight),dt\$.

Carry out the substitution. "Miraculously," our integral simplifies come \$int fracdtt\$.

re-superstructure
mention
monitor
answer Aug 5 "12 at 15:26

André NicolasAndré Nicolas
\$endgroup\$
5
\$egingroup\$
Put \$x= an y\$, so that \$dx=sec^2y dy\$ and \$sqrt1+x^2=sec y\$

\$\$int frac1sqrt1+x^2 dx\$\$

\$\$= int fracsec^2y dysec y\$\$

\$\$=int sec y, dy\$\$

which evaluates to \$displaystyleln|sec y+ an y|+ C\$ , applying the typical formula whose proof is here and \$C\$ is one indeterminate constant for any type of indefinite integral.

\$\$=ln|sqrt1+x^2+x| + C\$\$

We deserve to substitute \$x\$ v \$a sec y\$ because that \$sqrtx^2-a^2\$, and also with \$a sin y\$ for \$sqrta^2-x^2\$

share
cite
monitor
edited Aug 5 "12 in ~ 14:37
reply Aug 5 "12 in ~ 14:05

laboratory bhattacharjeelab bhattacharjee
\$endgroup\$
include a comment |
3
\$egingroup\$
\$\$A=intfrac1sqrt<>1+x^2\$\$

Let, \$x = an heta\$

substitute, \$x\$, \$dx\$

\$\$A=intsec hetaleft(fracsec heta + an hetasec heta + an heta ight)d heta\$\$

\$\$A=intleft(fracsec^2 heta + sec heta an hetasec heta + an heta ight)d heta\$\$

Let, \$(sec heta + an heta) = u\$

\$(sec^2 heta + sec heta an heta)d heta = du\$

\$\$A=intfracduu\$\$

\$\$A=lnu+c\$\$

\$\$A=lnvertsec heta + an hetavert+c\$\$

\$\$A=lnvertsqrt<>1+ an^2 heta + an hetavert+c\$\$

\$A=lnvertsqrt<>1+x^2 + xvert+c\$, whereby \$c\$ is a constant

re-superstructure
point out
follow
reply Aug 5 "12 at 17:37
HOLYBIBLETHEHOLYBIBLETHE
\$endgroup\$

Thanks because that contributing an answer to music-from-a.com Stack Exchange!

But avoid

Asking because that help, clarification, or responding to various other answers.Making statements based upon opinion; back them up with recommendations or an individual experience.

Use music-from-a.comJax to format equations. music-from-a.comJax reference.

To find out more, check out our tips on writing good answers.

See more: T&Amp;T Repairables Spencer In E, At&T Official Site

Draft saved

authorize up utilizing Email and Password
submit

### Post as a guest

name
email Required, however never shown

### Post together a guest

surname
email

Required, but never shown

Featured top top Meta
1
Solving the integral \$intfrac1sqrtx^2+1,dx\$
4
How to incorporate \$int dx over sqrt1 + x^2\$
6
The integral \$intfrac2(2y^2+1)(y^2+1)^0.5 dy\$
1
Evaluate the integral \$int fraccos(x)sqrt1+sin^2(x) , dx\$
1
Compute \$int_0^1frac sqrtx(x+3)sqrtx+3dx.\$
connected
4
Integration the \$ int fracdxx^2sqrtx^2 + 9 \$ using trigonometric substitution
1
combine using substitution
3
how to combine \$frac1xsqrt1+x^2\$ utilizing substitution?
0
Integration making use of hyperbolic substitution
0
resolve \$ intfracsqrtx-1xdx\$ by utilizing substitution
2
integrate \$x^2sin(2x)\$ utilizing \$u\$-substitution
1
just how to incorporate \$int frac816-e^4x music-from-a.comrm dx\$ using trigonometric substitution?
9
combine via substitution: \$x^2sqrtx^2+1;dx\$
warm Network concerns much more hot inquiries

inquiry feed

music-from-a.comematics
agency
ridge Exchange Network
site style / logo design © 2021 stack Exchange Inc; user contributions license is granted under cc by-sa. Rev2021.12.22.41046

music-from-a.comematics ridge Exchange works finest with JavaScript permitted