(4) Derive the composite trapazoidal rule and composite simpson's rule from simple trapazoidal and simple simpson's rule
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Ref Lecture notes p.9-3
Show Simple Trapazoidal rulep.7-1 
Composite Trapazoidal rule p.7-1
and
Show Simple Simpson's rule p.7-2 Composite Simpson's rulep.7-2
Composite Trapezoidal rule
From Simple trapezoidal rule we have ,
![{\displaystyle I_{1}={\frac {h}{2}}[f_{\text{o}}+f_{1}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb57678700de1c99a0ea3cb31ce27da18f35a59d)
![{\displaystyle =h[{\frac {1}{2}}f_{\text{o}}+{\frac {1}{2}}f_{1}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0a71f2640e66a8e946bbb61fddf77ccce672fef)
similarly we have
![{\displaystyle I_{2}=h[{\frac {1}{2}}f_{1}+{\frac {1}{2}}f_{2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da6c17d10c6b7a6056a3c2135f71addb0cc1ce2a)
![{\displaystyle I_{3}=h[{\frac {1}{2}}f_{2}+{\frac {1}{2}}f_{3}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c09456e088ae45a9f72d11c6969c8debfe184fd3)
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![{\displaystyle I_{\text{n-1}}=h[{\frac {1}{2}}f_{\text{n-2}}+{\frac {1}{2}}f_{\text{n-1}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af64845ea9e84c502433d45a75ecbcf1c2c56a8a)
![{\displaystyle I_{\text{n}}=h[{\frac {1}{2}}f_{\text{n-1}}+{\frac {1}{2}}f_{\text{n}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0524cede45d21d11d77b34548bb85a283a55ba1)
Summation of all of above expression gives
![{\displaystyle I_{\text{n}}=h[{\frac {1}{2}}f_{\text{o}}+({\frac {1}{2}}f_{1}+{\frac {1}{2}}f_{1})+({\frac {1}{2}}f_{2}+{\frac {1}{2}}f_{2})+.....+({\frac {1}{2}}f_{\text{n-1}}+{\frac {1}{2}}f_{\text{n-1}})+{\frac {1}{2}}f_{\text{n}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13e0713acca77d4a362720cab9e806d6d513130d)
![{\displaystyle =h[{\frac {1}{2}}f_{\text{o}}+f_{1}+f_{2}+.......+f_{\text{n-1}}+{\frac {1}{2}}f_{\text{n}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46eb305af82ef60fd0e2a0151bfd218a1ce63be7)
Hence Proved..
Composite Simpson's Rule
From Simple Simpson's rule we obtain ,
![{\displaystyle I_{2}={\frac {h}{3}}[f_{\text{o}}+4f_{1}+f_{2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82df5d70d6bcd53cba8537006c8106ee699b62bd)
where 
Similarly
![{\displaystyle I_{4}={\frac {h}{3}}[f_{2}+4f_{3}+f_{4}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ec7a8f3fa21557a0014ec2d47beafb64b249c6e)
![{\displaystyle I_{6}={\frac {h}{3}}[f_{4}+4f_{5}+f_{6}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21b5558161c82e7f5fbcc7b64965473df431ccac)
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![{\displaystyle I_{\text{n-2}}={\frac {h}{3}}[f_{\text{n-4}}+4f_{\text{n-3}}+f_{\text{n-2}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8092374d3d6febff8db3e556257277fdefdcca5)
![{\displaystyle I_{\text{n}}={\frac {h}{3}}[f_{\text{n-2}}+4f_{\text{n-1}}+f_{\text{n}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bfa9d54764d1575b96837a3ed792a89b7f041de)
After Summation of above terms we obtain
![{\displaystyle I_{\text{n}}={\frac {h}{3}}[f_{\text{o}}+4f_{1}+(f_{2}+f_{2})+4f_{\text{3}}+(f_{4}+f_{4})+4f_{5}+f_{6}.......+(f_{\text{n-2}}+f_{\text{n-2}})+4f_{\text{n-1}}+f_{\text{n}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e218952cba35591e72a0bafb61cdc708dbaa5a99)
![{\displaystyle I_{\text{n}}={\frac {h}{3}}[f_{\text{o}}+4f_{1}+2f_{2}+4f_{\text{3}}+2f_{4}+4f_{5}+f_{6}.......+2f_{\text{n-2}}+4f_{\text{n-1}}+f_{\text{n}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f34abb20caca0a45d0ab6f38ed17287445d45ed9)
where n = 2k and k = 1,2,3,4.....
Hence Proved
![{\displaystyle e^{(3)}(t)=-{\frac {t}{3}}[F^{(3)}(t)-F^{(3)}(-t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d4bf86aa7082193ee20a3839f8791a5ba54e6da)
Ref Lecture p.15-2
Prove that
where
![{\displaystyle e(t)=\int _{-t}^{t}f(x(t))\,dt-{\frac {t}{3}}[F(-t)+4F(0)+F(t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36587ed4aa8230c0a876ce144d3a3deab782fcc6)
![{\displaystyle e(t)=\int _{-t}^{k}f(x(t))\,dt+\int _{k}^{t}f(x(t))\,dt-{\frac {t}{3}}[F(-t)+4F(0)+F(t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e3f32446166978cc3fd84fdb5f5ae4d6942027f)
![{\displaystyle e^{(1)}(t)=(F(-t)+F(t))-{\frac {t}{3}}[-F^{1}(-t)+F^{1}(t)]-{\frac {1}{3}}[F(-t)+4F(0)+F(t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9d11e309b8fe42a4a75d7fba6b86f5b38aab5d5)
![{\displaystyle e^{(2)}(t)=(-F^{1}(-t)+F^{1}(t))-{\frac {t}{3}}[F^{2}(-t)+F^{2}(t)]-{\frac {1}{3}}[-F^{1}(-t)+F^{1}(t)]-{\frac {1}{3}}[-F^{1}(-t)+F^{1}(t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac5e8eb2b4f430930cfda6a26f56f196f8f57f6)
![{\displaystyle e^{(2)}(t)=(-F^{1}(-t)+F^{1}(t))-{\frac {t}{3}}[F^{2}(-t)+F^{2}(t)]-{\frac {2}{3}}[-F^{1}(-t)+F^{1}(t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee3fcaa7ab1097dfce40cca0b6f7ee7538914f9b)
![{\displaystyle e^{(3)}(t)=(F^{2}(-t)+F^{2}(t))-{\frac {t}{3}}[-F^{3}(-t)+F^{3}(t)]-{\frac {1}{3}}[F^{2}(-t)+F^{2}(t)]-{\frac {2}{3}}[F^{2}(-t)+F^{2}(t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9156a3583b70429b3440175e98a5bd6691b81574)
![{\displaystyle e^{(3)}(t)=(F^{2}(-t)+F^{2}(t))-{\frac {t}{3}}[-F^{3}(-t)+F^{3}(t)]-(F^{2}(-t)+F^{2}(t))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11c5a60e76f6bc190a5e603a948c98c41e093e79)
![{\displaystyle e^{(3)}(t)=-{\frac {t}{3}}[-F^{3}(-t)+F^{3}(t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b328cb8d34196ea7dbf7d150165296e617e3018)
![{\displaystyle e^{(3)}(t)=-{\frac {t}{3}}[F^{3}(t)-F^{3}(-t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5f05c08ae7a3e13af1125fec060f60a7835f0a8)
Hence Proved ..
(12) Show Derivation
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Ref: Lecture Notes p.15-1
Show derivation that
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From (4) in p.14-2, we can write down the expression of 
Given :
f(x(t)) = F(t)
Let there exists 
![{\displaystyle \alpha ^{(1)}(t)={\frac {d}{dt}}[g(t)-g(-t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35465cce48e874e60dfa7686d4808d339ba147b2)
but
Hence Proved ,
