Better way to solve integral











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2
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I have the following integral



$$int_{0}^{pi / 2} mathrm{d} x ,, frac{cos^{3}(x/2) - cos^{4} (x)}{sin^{2} (x)}$$



My current solution is to use
$$v = tan left(frac{x}{4}right)$$
to obtain a rational function of $v$, and integrate this.



Is there a more practical / clever way of doing it?










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  • Where did you see this integral?
    – Frpzzd
    2 days ago










  • Comes up in my work. I like the Weierstrass trick as it is a blanket method, but am looking for a better way to approach it
    – Gordon
    2 days ago















up vote
2
down vote

favorite












I have the following integral



$$int_{0}^{pi / 2} mathrm{d} x ,, frac{cos^{3}(x/2) - cos^{4} (x)}{sin^{2} (x)}$$



My current solution is to use
$$v = tan left(frac{x}{4}right)$$
to obtain a rational function of $v$, and integrate this.



Is there a more practical / clever way of doing it?










share|cite|improve this question






















  • Where did you see this integral?
    – Frpzzd
    2 days ago










  • Comes up in my work. I like the Weierstrass trick as it is a blanket method, but am looking for a better way to approach it
    – Gordon
    2 days ago













up vote
2
down vote

favorite









up vote
2
down vote

favorite











I have the following integral



$$int_{0}^{pi / 2} mathrm{d} x ,, frac{cos^{3}(x/2) - cos^{4} (x)}{sin^{2} (x)}$$



My current solution is to use
$$v = tan left(frac{x}{4}right)$$
to obtain a rational function of $v$, and integrate this.



Is there a more practical / clever way of doing it?










share|cite|improve this question













I have the following integral



$$int_{0}^{pi / 2} mathrm{d} x ,, frac{cos^{3}(x/2) - cos^{4} (x)}{sin^{2} (x)}$$



My current solution is to use
$$v = tan left(frac{x}{4}right)$$
to obtain a rational function of $v$, and integrate this.



Is there a more practical / clever way of doing it?







integration






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 2 days ago









Gordon

364




364












  • Where did you see this integral?
    – Frpzzd
    2 days ago










  • Comes up in my work. I like the Weierstrass trick as it is a blanket method, but am looking for a better way to approach it
    – Gordon
    2 days ago


















  • Where did you see this integral?
    – Frpzzd
    2 days ago










  • Comes up in my work. I like the Weierstrass trick as it is a blanket method, but am looking for a better way to approach it
    – Gordon
    2 days ago
















Where did you see this integral?
– Frpzzd
2 days ago




Where did you see this integral?
– Frpzzd
2 days ago












Comes up in my work. I like the Weierstrass trick as it is a blanket method, but am looking for a better way to approach it
– Gordon
2 days ago




Comes up in my work. I like the Weierstrass trick as it is a blanket method, but am looking for a better way to approach it
– Gordon
2 days ago










3 Answers
3






active

oldest

votes

















up vote
0
down vote













While the substitution $v=tan(x/4)$ would work, I don't think it's a good method in this case, which just requires elementary antiderivatives.



First, split the fraction and notice that
$$
intfrac{cos^4x}{sin^2x},dx=intleft(frac{1}{sin^2x}-2+sin^2xright),dx
=-cot x-2x+frac{1}{2}(x-sin xcos x)
$$

This leaves the other piece:
$$
intfrac{cos^3(x/2)}{sin^2x},dx=[t=x/2]=2intfrac{cos^3t}{sin^22t},dt
=frac{1}{2}intfrac{cos t}{sin^2t},dt=-frac{1}{2sin t}
$$

Thus the integral is
$$
-frac{1}{2sin(x/2)}+cot x+2x-frac{1}{2}(x-sin xcos x)
$$

You can also note that
$$
cot x-frac{1}{2sin(x/2)}=frac{cos x}{2sin(x/2)cos(x/2)}-frac{1}{2sin(x/2)}=
frac{(cos(x/2)-1)(2cos(x/2)+1)}{2sin(x/2)}
$$

which has limit $0$ for $xto0$.






share|cite|improve this answer




























    up vote
    0
    down vote













    $$frac{cos^3dfrac x2}{sin^2x}=frac{cosdfrac x2}{2sin^2dfrac x2}$$ is immediately integrable.



    And in



    $$frac{cos^4x}{sin^2x}=frac1{sin^2x}-2+sin^2x,$$ the first two terms are also immediate, and



    $$sin^2x=frac{1-cos2x}2.$$






    share|cite|improve this answer




























      up vote
      0
      down vote













      $$I=intfrac{cos^3(frac x2)-cos^4x}{sin^2x}mathrm{d}x$$
      $$I=intfrac{cos^3(frac x2)}{sin^2x}mathrm{d}x-intfrac{cos^4x}{sin^2x}mathrm{d}x$$
      $$I=big(I_1-I_2big)bigg|_0^{pi/2}$$





      $$I_1=intfrac{cos^3(frac x2)}{sin^2x}mathrm{d}x$$
      $u=x/2$:
      $$I_1=2intfrac{cos^3u}{sin^22u}mathrm{d}u$$
      $$I_1=2intfrac{cos^3u}{4sin^2ucos^2u}mathrm{d}u$$
      $$I_1=frac12intfrac{cos u}{sin^2u}mathrm{d}u$$
      $t=sin u$:
      $$I_1=frac12intfrac{mathrm{d}t}{t^2}$$
      $$I_1=-frac1{2t}$$
      $$I_1=-frac1{2sin(x/2)}$$





      $$I_2=intfrac{cos^4x}{sin^2x}mathrm{d}x$$
      $$I_2=intfrac{cos^2x(1-sin^2x)}{sin^2x}mathrm{d}x$$
      $$I_2=intcot^2x mathrm{d}x-intcos^2x mathrm{d}x$$
      $$I_2=-bigg(cot x+frac12cos x,sin x+frac32xbigg)$$





      $$I=bigg(-frac12csc(x/2)+cot x+frac12cos xsin x+frac32xbigg)bigg|_0^{pi/2}$$
      $$I=frac{3pi-2sqrt{2}}4$$






      share|cite|improve this answer





















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        3 Answers
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        active

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        3 Answers
        3






        active

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        active

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        active

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        up vote
        0
        down vote













        While the substitution $v=tan(x/4)$ would work, I don't think it's a good method in this case, which just requires elementary antiderivatives.



        First, split the fraction and notice that
        $$
        intfrac{cos^4x}{sin^2x},dx=intleft(frac{1}{sin^2x}-2+sin^2xright),dx
        =-cot x-2x+frac{1}{2}(x-sin xcos x)
        $$

        This leaves the other piece:
        $$
        intfrac{cos^3(x/2)}{sin^2x},dx=[t=x/2]=2intfrac{cos^3t}{sin^22t},dt
        =frac{1}{2}intfrac{cos t}{sin^2t},dt=-frac{1}{2sin t}
        $$

        Thus the integral is
        $$
        -frac{1}{2sin(x/2)}+cot x+2x-frac{1}{2}(x-sin xcos x)
        $$

        You can also note that
        $$
        cot x-frac{1}{2sin(x/2)}=frac{cos x}{2sin(x/2)cos(x/2)}-frac{1}{2sin(x/2)}=
        frac{(cos(x/2)-1)(2cos(x/2)+1)}{2sin(x/2)}
        $$

        which has limit $0$ for $xto0$.






        share|cite|improve this answer

























          up vote
          0
          down vote













          While the substitution $v=tan(x/4)$ would work, I don't think it's a good method in this case, which just requires elementary antiderivatives.



          First, split the fraction and notice that
          $$
          intfrac{cos^4x}{sin^2x},dx=intleft(frac{1}{sin^2x}-2+sin^2xright),dx
          =-cot x-2x+frac{1}{2}(x-sin xcos x)
          $$

          This leaves the other piece:
          $$
          intfrac{cos^3(x/2)}{sin^2x},dx=[t=x/2]=2intfrac{cos^3t}{sin^22t},dt
          =frac{1}{2}intfrac{cos t}{sin^2t},dt=-frac{1}{2sin t}
          $$

          Thus the integral is
          $$
          -frac{1}{2sin(x/2)}+cot x+2x-frac{1}{2}(x-sin xcos x)
          $$

          You can also note that
          $$
          cot x-frac{1}{2sin(x/2)}=frac{cos x}{2sin(x/2)cos(x/2)}-frac{1}{2sin(x/2)}=
          frac{(cos(x/2)-1)(2cos(x/2)+1)}{2sin(x/2)}
          $$

          which has limit $0$ for $xto0$.






          share|cite|improve this answer























            up vote
            0
            down vote










            up vote
            0
            down vote









            While the substitution $v=tan(x/4)$ would work, I don't think it's a good method in this case, which just requires elementary antiderivatives.



            First, split the fraction and notice that
            $$
            intfrac{cos^4x}{sin^2x},dx=intleft(frac{1}{sin^2x}-2+sin^2xright),dx
            =-cot x-2x+frac{1}{2}(x-sin xcos x)
            $$

            This leaves the other piece:
            $$
            intfrac{cos^3(x/2)}{sin^2x},dx=[t=x/2]=2intfrac{cos^3t}{sin^22t},dt
            =frac{1}{2}intfrac{cos t}{sin^2t},dt=-frac{1}{2sin t}
            $$

            Thus the integral is
            $$
            -frac{1}{2sin(x/2)}+cot x+2x-frac{1}{2}(x-sin xcos x)
            $$

            You can also note that
            $$
            cot x-frac{1}{2sin(x/2)}=frac{cos x}{2sin(x/2)cos(x/2)}-frac{1}{2sin(x/2)}=
            frac{(cos(x/2)-1)(2cos(x/2)+1)}{2sin(x/2)}
            $$

            which has limit $0$ for $xto0$.






            share|cite|improve this answer












            While the substitution $v=tan(x/4)$ would work, I don't think it's a good method in this case, which just requires elementary antiderivatives.



            First, split the fraction and notice that
            $$
            intfrac{cos^4x}{sin^2x},dx=intleft(frac{1}{sin^2x}-2+sin^2xright),dx
            =-cot x-2x+frac{1}{2}(x-sin xcos x)
            $$

            This leaves the other piece:
            $$
            intfrac{cos^3(x/2)}{sin^2x},dx=[t=x/2]=2intfrac{cos^3t}{sin^22t},dt
            =frac{1}{2}intfrac{cos t}{sin^2t},dt=-frac{1}{2sin t}
            $$

            Thus the integral is
            $$
            -frac{1}{2sin(x/2)}+cot x+2x-frac{1}{2}(x-sin xcos x)
            $$

            You can also note that
            $$
            cot x-frac{1}{2sin(x/2)}=frac{cos x}{2sin(x/2)cos(x/2)}-frac{1}{2sin(x/2)}=
            frac{(cos(x/2)-1)(2cos(x/2)+1)}{2sin(x/2)}
            $$

            which has limit $0$ for $xto0$.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 2 days ago









            egreg

            173k1383198




            173k1383198






















                up vote
                0
                down vote













                $$frac{cos^3dfrac x2}{sin^2x}=frac{cosdfrac x2}{2sin^2dfrac x2}$$ is immediately integrable.



                And in



                $$frac{cos^4x}{sin^2x}=frac1{sin^2x}-2+sin^2x,$$ the first two terms are also immediate, and



                $$sin^2x=frac{1-cos2x}2.$$






                share|cite|improve this answer

























                  up vote
                  0
                  down vote













                  $$frac{cos^3dfrac x2}{sin^2x}=frac{cosdfrac x2}{2sin^2dfrac x2}$$ is immediately integrable.



                  And in



                  $$frac{cos^4x}{sin^2x}=frac1{sin^2x}-2+sin^2x,$$ the first two terms are also immediate, and



                  $$sin^2x=frac{1-cos2x}2.$$






                  share|cite|improve this answer























                    up vote
                    0
                    down vote










                    up vote
                    0
                    down vote









                    $$frac{cos^3dfrac x2}{sin^2x}=frac{cosdfrac x2}{2sin^2dfrac x2}$$ is immediately integrable.



                    And in



                    $$frac{cos^4x}{sin^2x}=frac1{sin^2x}-2+sin^2x,$$ the first two terms are also immediate, and



                    $$sin^2x=frac{1-cos2x}2.$$






                    share|cite|improve this answer












                    $$frac{cos^3dfrac x2}{sin^2x}=frac{cosdfrac x2}{2sin^2dfrac x2}$$ is immediately integrable.



                    And in



                    $$frac{cos^4x}{sin^2x}=frac1{sin^2x}-2+sin^2x,$$ the first two terms are also immediate, and



                    $$sin^2x=frac{1-cos2x}2.$$







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 2 days ago









                    Yves Daoust

                    121k668216




                    121k668216






















                        up vote
                        0
                        down vote













                        $$I=intfrac{cos^3(frac x2)-cos^4x}{sin^2x}mathrm{d}x$$
                        $$I=intfrac{cos^3(frac x2)}{sin^2x}mathrm{d}x-intfrac{cos^4x}{sin^2x}mathrm{d}x$$
                        $$I=big(I_1-I_2big)bigg|_0^{pi/2}$$





                        $$I_1=intfrac{cos^3(frac x2)}{sin^2x}mathrm{d}x$$
                        $u=x/2$:
                        $$I_1=2intfrac{cos^3u}{sin^22u}mathrm{d}u$$
                        $$I_1=2intfrac{cos^3u}{4sin^2ucos^2u}mathrm{d}u$$
                        $$I_1=frac12intfrac{cos u}{sin^2u}mathrm{d}u$$
                        $t=sin u$:
                        $$I_1=frac12intfrac{mathrm{d}t}{t^2}$$
                        $$I_1=-frac1{2t}$$
                        $$I_1=-frac1{2sin(x/2)}$$





                        $$I_2=intfrac{cos^4x}{sin^2x}mathrm{d}x$$
                        $$I_2=intfrac{cos^2x(1-sin^2x)}{sin^2x}mathrm{d}x$$
                        $$I_2=intcot^2x mathrm{d}x-intcos^2x mathrm{d}x$$
                        $$I_2=-bigg(cot x+frac12cos x,sin x+frac32xbigg)$$





                        $$I=bigg(-frac12csc(x/2)+cot x+frac12cos xsin x+frac32xbigg)bigg|_0^{pi/2}$$
                        $$I=frac{3pi-2sqrt{2}}4$$






                        share|cite|improve this answer

























                          up vote
                          0
                          down vote













                          $$I=intfrac{cos^3(frac x2)-cos^4x}{sin^2x}mathrm{d}x$$
                          $$I=intfrac{cos^3(frac x2)}{sin^2x}mathrm{d}x-intfrac{cos^4x}{sin^2x}mathrm{d}x$$
                          $$I=big(I_1-I_2big)bigg|_0^{pi/2}$$





                          $$I_1=intfrac{cos^3(frac x2)}{sin^2x}mathrm{d}x$$
                          $u=x/2$:
                          $$I_1=2intfrac{cos^3u}{sin^22u}mathrm{d}u$$
                          $$I_1=2intfrac{cos^3u}{4sin^2ucos^2u}mathrm{d}u$$
                          $$I_1=frac12intfrac{cos u}{sin^2u}mathrm{d}u$$
                          $t=sin u$:
                          $$I_1=frac12intfrac{mathrm{d}t}{t^2}$$
                          $$I_1=-frac1{2t}$$
                          $$I_1=-frac1{2sin(x/2)}$$





                          $$I_2=intfrac{cos^4x}{sin^2x}mathrm{d}x$$
                          $$I_2=intfrac{cos^2x(1-sin^2x)}{sin^2x}mathrm{d}x$$
                          $$I_2=intcot^2x mathrm{d}x-intcos^2x mathrm{d}x$$
                          $$I_2=-bigg(cot x+frac12cos x,sin x+frac32xbigg)$$





                          $$I=bigg(-frac12csc(x/2)+cot x+frac12cos xsin x+frac32xbigg)bigg|_0^{pi/2}$$
                          $$I=frac{3pi-2sqrt{2}}4$$






                          share|cite|improve this answer























                            up vote
                            0
                            down vote










                            up vote
                            0
                            down vote









                            $$I=intfrac{cos^3(frac x2)-cos^4x}{sin^2x}mathrm{d}x$$
                            $$I=intfrac{cos^3(frac x2)}{sin^2x}mathrm{d}x-intfrac{cos^4x}{sin^2x}mathrm{d}x$$
                            $$I=big(I_1-I_2big)bigg|_0^{pi/2}$$





                            $$I_1=intfrac{cos^3(frac x2)}{sin^2x}mathrm{d}x$$
                            $u=x/2$:
                            $$I_1=2intfrac{cos^3u}{sin^22u}mathrm{d}u$$
                            $$I_1=2intfrac{cos^3u}{4sin^2ucos^2u}mathrm{d}u$$
                            $$I_1=frac12intfrac{cos u}{sin^2u}mathrm{d}u$$
                            $t=sin u$:
                            $$I_1=frac12intfrac{mathrm{d}t}{t^2}$$
                            $$I_1=-frac1{2t}$$
                            $$I_1=-frac1{2sin(x/2)}$$





                            $$I_2=intfrac{cos^4x}{sin^2x}mathrm{d}x$$
                            $$I_2=intfrac{cos^2x(1-sin^2x)}{sin^2x}mathrm{d}x$$
                            $$I_2=intcot^2x mathrm{d}x-intcos^2x mathrm{d}x$$
                            $$I_2=-bigg(cot x+frac12cos x,sin x+frac32xbigg)$$





                            $$I=bigg(-frac12csc(x/2)+cot x+frac12cos xsin x+frac32xbigg)bigg|_0^{pi/2}$$
                            $$I=frac{3pi-2sqrt{2}}4$$






                            share|cite|improve this answer












                            $$I=intfrac{cos^3(frac x2)-cos^4x}{sin^2x}mathrm{d}x$$
                            $$I=intfrac{cos^3(frac x2)}{sin^2x}mathrm{d}x-intfrac{cos^4x}{sin^2x}mathrm{d}x$$
                            $$I=big(I_1-I_2big)bigg|_0^{pi/2}$$





                            $$I_1=intfrac{cos^3(frac x2)}{sin^2x}mathrm{d}x$$
                            $u=x/2$:
                            $$I_1=2intfrac{cos^3u}{sin^22u}mathrm{d}u$$
                            $$I_1=2intfrac{cos^3u}{4sin^2ucos^2u}mathrm{d}u$$
                            $$I_1=frac12intfrac{cos u}{sin^2u}mathrm{d}u$$
                            $t=sin u$:
                            $$I_1=frac12intfrac{mathrm{d}t}{t^2}$$
                            $$I_1=-frac1{2t}$$
                            $$I_1=-frac1{2sin(x/2)}$$





                            $$I_2=intfrac{cos^4x}{sin^2x}mathrm{d}x$$
                            $$I_2=intfrac{cos^2x(1-sin^2x)}{sin^2x}mathrm{d}x$$
                            $$I_2=intcot^2x mathrm{d}x-intcos^2x mathrm{d}x$$
                            $$I_2=-bigg(cot x+frac12cos x,sin x+frac32xbigg)$$





                            $$I=bigg(-frac12csc(x/2)+cot x+frac12cos xsin x+frac32xbigg)bigg|_0^{pi/2}$$
                            $$I=frac{3pi-2sqrt{2}}4$$







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered 2 days ago









                            clathratus

                            1,839219




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