Integration by Parts only of $sqrt{1-u^2}$












0












$begingroup$


I am trying to integrate the function:
$$f(x)=sqrt{1-u^2}$$
I was using integration by parts to attack the problem, and it was:
$$intsqrt{1-u^2}du$$
I set $g=sqrt{1-u^2}$ and $dv=du$



Thus leading me to get:
$$usqrt{1-u^2}+intfrac{u^2}{sqrt{1-u^2}}$$



from there I set $g=u$, and $dv=frac{u}{sqrt{1-u^2}}du$



$$I=usqrt{1-u^2}-usqrt{1-u^2}-I$$



I somehow lose the inverse sine portion of the answer.



Just using integration by parts is there a way I can get the right answer.






calculus integration trigonometric-integrals







share|cite|improve this question











$endgroup$












  • $begingroup$
    Hint: Let $u = sinx$ and $du = cosx dx$.
    $endgroup$
    – T. Fo
    Jan 8 at 17:18










  • $begingroup$
    You forgot the sign in the integration by parts formula it should be a negative integral. I don't think you can solve this integral using integration by parts...
    $endgroup$
    – Peter Foreman
    Jan 8 at 17:18










  • $begingroup$
    I believe one usually uses trig substitution for such things, try $u = sin x$
    $endgroup$
    – gt6989b
    Jan 8 at 17:18










  • $begingroup$
    Oh I misread, my hint is for substitution. I'm not sure about IbP.
    $endgroup$
    – T. Fo
    Jan 8 at 17:19










  • $begingroup$
    @PeterForeman Thanks I did it the other way already thanks though, I looked up another similar question.
    $endgroup$
    – EnlightenedFunky
    Jan 8 at 17:19
















0












$begingroup$


I am trying to integrate the function:
$$f(x)=sqrt{1-u^2}$$
I was using integration by parts to attack the problem, and it was:
$$intsqrt{1-u^2}du$$
I set $g=sqrt{1-u^2}$ and $dv=du$



Thus leading me to get:
$$usqrt{1-u^2}+intfrac{u^2}{sqrt{1-u^2}}$$



from there I set $g=u$, and $dv=frac{u}{sqrt{1-u^2}}du$



$$I=usqrt{1-u^2}-usqrt{1-u^2}-I$$



I somehow lose the inverse sine portion of the answer.



Just using integration by parts is there a way I can get the right answer.






calculus integration trigonometric-integrals







share|cite|improve this question











$endgroup$












  • $begingroup$
    Hint: Let $u = sinx$ and $du = cosx dx$.
    $endgroup$
    – T. Fo
    Jan 8 at 17:18










  • $begingroup$
    You forgot the sign in the integration by parts formula it should be a negative integral. I don't think you can solve this integral using integration by parts...
    $endgroup$
    – Peter Foreman
    Jan 8 at 17:18










  • $begingroup$
    I believe one usually uses trig substitution for such things, try $u = sin x$
    $endgroup$
    – gt6989b
    Jan 8 at 17:18










  • $begingroup$
    Oh I misread, my hint is for substitution. I'm not sure about IbP.
    $endgroup$
    – T. Fo
    Jan 8 at 17:19










  • $begingroup$
    @PeterForeman Thanks I did it the other way already thanks though, I looked up another similar question.
    $endgroup$
    – EnlightenedFunky
    Jan 8 at 17:19














0












0








0





$begingroup$


I am trying to integrate the function:
$$f(x)=sqrt{1-u^2}$$
I was using integration by parts to attack the problem, and it was:
$$intsqrt{1-u^2}du$$
I set $g=sqrt{1-u^2}$ and $dv=du$



Thus leading me to get:
$$usqrt{1-u^2}+intfrac{u^2}{sqrt{1-u^2}}$$



from there I set $g=u$, and $dv=frac{u}{sqrt{1-u^2}}du$



$$I=usqrt{1-u^2}-usqrt{1-u^2}-I$$



I somehow lose the inverse sine portion of the answer.



Just using integration by parts is there a way I can get the right answer.






calculus integration trigonometric-integrals







share|cite|improve this question











$endgroup$




I am trying to integrate the function:
$$f(x)=sqrt{1-u^2}$$
I was using integration by parts to attack the problem, and it was:
$$intsqrt{1-u^2}du$$
I set $g=sqrt{1-u^2}$ and $dv=du$



Thus leading me to get:
$$usqrt{1-u^2}+intfrac{u^2}{sqrt{1-u^2}}$$



from there I set $g=u$, and $dv=frac{u}{sqrt{1-u^2}}du$



$$I=usqrt{1-u^2}-usqrt{1-u^2}-I$$



I somehow lose the inverse sine portion of the answer.



Just using integration by parts is there a way I can get the right answer.






calculus integration trigonometric-integrals




calculus integration trigonometric-integrals






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 8 at 17:19









gt6989b

33.9k22455




33.9k22455










asked Jan 8 at 17:14









EnlightenedFunkyEnlightenedFunky

7471822




7471822












  • $begingroup$
    Hint: Let $u = sinx$ and $du = cosx dx$.
    $endgroup$
    – T. Fo
    Jan 8 at 17:18










  • $begingroup$
    You forgot the sign in the integration by parts formula it should be a negative integral. I don't think you can solve this integral using integration by parts...
    $endgroup$
    – Peter Foreman
    Jan 8 at 17:18










  • $begingroup$
    I believe one usually uses trig substitution for such things, try $u = sin x$
    $endgroup$
    – gt6989b
    Jan 8 at 17:18










  • $begingroup$
    Oh I misread, my hint is for substitution. I'm not sure about IbP.
    $endgroup$
    – T. Fo
    Jan 8 at 17:19










  • $begingroup$
    @PeterForeman Thanks I did it the other way already thanks though, I looked up another similar question.
    $endgroup$
    – EnlightenedFunky
    Jan 8 at 17:19


















  • $begingroup$
    Hint: Let $u = sinx$ and $du = cosx dx$.
    $endgroup$
    – T. Fo
    Jan 8 at 17:18










  • $begingroup$
    You forgot the sign in the integration by parts formula it should be a negative integral. I don't think you can solve this integral using integration by parts...
    $endgroup$
    – Peter Foreman
    Jan 8 at 17:18










  • $begingroup$
    I believe one usually uses trig substitution for such things, try $u = sin x$
    $endgroup$
    – gt6989b
    Jan 8 at 17:18










  • $begingroup$
    Oh I misread, my hint is for substitution. I'm not sure about IbP.
    $endgroup$
    – T. Fo
    Jan 8 at 17:19










  • $begingroup$
    @PeterForeman Thanks I did it the other way already thanks though, I looked up another similar question.
    $endgroup$
    – EnlightenedFunky
    Jan 8 at 17:19
















$begingroup$
Hint: Let $u = sinx$ and $du = cosx dx$.
$endgroup$
– T. Fo
Jan 8 at 17:18




$begingroup$
Hint: Let $u = sinx$ and $du = cosx dx$.
$endgroup$
– T. Fo
Jan 8 at 17:18












$begingroup$
You forgot the sign in the integration by parts formula it should be a negative integral. I don't think you can solve this integral using integration by parts...
$endgroup$
– Peter Foreman
Jan 8 at 17:18




$begingroup$
You forgot the sign in the integration by parts formula it should be a negative integral. I don't think you can solve this integral using integration by parts...
$endgroup$
– Peter Foreman
Jan 8 at 17:18












$begingroup$
I believe one usually uses trig substitution for such things, try $u = sin x$
$endgroup$
– gt6989b
Jan 8 at 17:18




$begingroup$
I believe one usually uses trig substitution for such things, try $u = sin x$
$endgroup$
– gt6989b
Jan 8 at 17:18












$begingroup$
Oh I misread, my hint is for substitution. I'm not sure about IbP.
$endgroup$
– T. Fo
Jan 8 at 17:19




$begingroup$
Oh I misread, my hint is for substitution. I'm not sure about IbP.
$endgroup$
– T. Fo
Jan 8 at 17:19












$begingroup$
@PeterForeman Thanks I did it the other way already thanks though, I looked up another similar question.
$endgroup$
– EnlightenedFunky
Jan 8 at 17:19




$begingroup$
@PeterForeman Thanks I did it the other way already thanks though, I looked up another similar question.
$endgroup$
– EnlightenedFunky
Jan 8 at 17:19










1 Answer
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$begingroup$

Let $I = intsqrt{1-u^2}, du$.



begin{eqnarray*} I
&=& usqrt{1-u^2} + int frac{u^2}{sqrt{1-u^2}} , du \
&=& usqrt{1-u^2} - int frac{1-u^2 - 1}{sqrt{1-u^2}} , du \
&=& usqrt{1-u^2} - I +int frac{1}{sqrt{1-u^2}} , du \
&=& usqrt{1-u^2} - I + arcsin(u)\
end{eqnarray*}



It follows:
$$2I = usqrt{1-u^2} + arcsin(u) leftrightarrow I = frac{1}{2}left( usqrt{1-u^2} + arcsin(u)right) (+ C)$$






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    1 Answer
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    1 Answer
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    active

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    10












    $begingroup$

    Let $I = intsqrt{1-u^2}, du$.



    begin{eqnarray*} I
    &=& usqrt{1-u^2} + int frac{u^2}{sqrt{1-u^2}} , du \
    &=& usqrt{1-u^2} - int frac{1-u^2 - 1}{sqrt{1-u^2}} , du \
    &=& usqrt{1-u^2} - I +int frac{1}{sqrt{1-u^2}} , du \
    &=& usqrt{1-u^2} - I + arcsin(u)\
    end{eqnarray*}



    It follows:
    $$2I = usqrt{1-u^2} + arcsin(u) leftrightarrow I = frac{1}{2}left( usqrt{1-u^2} + arcsin(u)right) (+ C)$$






    share|cite|improve this answer









    $endgroup$


















      10












      $begingroup$

      Let $I = intsqrt{1-u^2}, du$.



      begin{eqnarray*} I
      &=& usqrt{1-u^2} + int frac{u^2}{sqrt{1-u^2}} , du \
      &=& usqrt{1-u^2} - int frac{1-u^2 - 1}{sqrt{1-u^2}} , du \
      &=& usqrt{1-u^2} - I +int frac{1}{sqrt{1-u^2}} , du \
      &=& usqrt{1-u^2} - I + arcsin(u)\
      end{eqnarray*}



      It follows:
      $$2I = usqrt{1-u^2} + arcsin(u) leftrightarrow I = frac{1}{2}left( usqrt{1-u^2} + arcsin(u)right) (+ C)$$






      share|cite|improve this answer









      $endgroup$
















        10












        10








        10





        $begingroup$

        Let $I = intsqrt{1-u^2}, du$.



        begin{eqnarray*} I
        &=& usqrt{1-u^2} + int frac{u^2}{sqrt{1-u^2}} , du \
        &=& usqrt{1-u^2} - int frac{1-u^2 - 1}{sqrt{1-u^2}} , du \
        &=& usqrt{1-u^2} - I +int frac{1}{sqrt{1-u^2}} , du \
        &=& usqrt{1-u^2} - I + arcsin(u)\
        end{eqnarray*}



        It follows:
        $$2I = usqrt{1-u^2} + arcsin(u) leftrightarrow I = frac{1}{2}left( usqrt{1-u^2} + arcsin(u)right) (+ C)$$






        share|cite|improve this answer









        $endgroup$



        Let $I = intsqrt{1-u^2}, du$.



        begin{eqnarray*} I
        &=& usqrt{1-u^2} + int frac{u^2}{sqrt{1-u^2}} , du \
        &=& usqrt{1-u^2} - int frac{1-u^2 - 1}{sqrt{1-u^2}} , du \
        &=& usqrt{1-u^2} - I +int frac{1}{sqrt{1-u^2}} , du \
        &=& usqrt{1-u^2} - I + arcsin(u)\
        end{eqnarray*}



        It follows:
        $$2I = usqrt{1-u^2} + arcsin(u) leftrightarrow I = frac{1}{2}left( usqrt{1-u^2} + arcsin(u)right) (+ C)$$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 8 at 17:28









        trancelocationtrancelocation

        10.5k1722




        10.5k1722






























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