CALCULUS PRESENTED BY GUO CHENG GUANG（05-53-56）--Four Methods to Integrate

Guess, how many formulae have been used for integral ∫(1/(sinx+cosx))dx? More than eighteen formulae.

A.Trigonometric identitis:

   a. Pythagorean trigonometric identity:

      1. (sinα)^2+(cosα)^2=1

   b. co-function identities:

      2. sinx=cos(π/2-x)

      3. cosx=sin(π/2-x)

   c. angle sum identity

      4. tan(α+β)=(tanα+tanβ)/(1-tanαtanβ)

   d. double-angle formulae

      5. cos2x=(cosx)^2-(sinx)^2=2(cosx)^2-1=1-2(sinx)^2

      6. sin2x=2sinxcosx

   e. sum-to-product identities

      7. sinα+sinβ= 2sin((α+β)/2)cos((α-β)/2)

      8. cosα-cosβ= 2cos((α+β)/2)cos((α-β)/2)

      9. sinα+sinβ= 2cos((α+β)/2)sin((α-β)/2)

      10. cosα-cosβ= -2sin((α+β)/2)sin((α-β)/2)

B. Algebra Formulas - Factoring formulas- Difference of squares:

      11. a^2-b^2=(a-b)(a+b)

C. Euler's formula

      12. e^(ix)=cosx+isinx

      13. sinx=(e^(ix)-e^(-ix))/2i

      14. cosx=(e^(ix)+e^(-ix))/2

 D. Logarithm rules

      15. ln(x/y)=lnx-lny

 E.The complex inverse trigonometric function

      16. arctanz=i/2(ln((1-iz)/(1+iz)))

 F. Basic Integration formulae:

      17. ∫(1/x)dx=lnx+c

      18. ∫(1/(x^2+y^2))dx=1/a(arctan(x/a))+c

Calculus--Indefinite Integrals (05-53-56,S/N: 321) Presented By Guo Cheng Guang (aged 11) and Guo Chengxi (aged 8) in English. Lessons In Mathematics and Science (Physics, Chemistry and Biology) From Primary to University Presented By Guo Cheng Guang and Guo Chengxi In English. Your comment or suggestion is very much appreciated. https://www.youtube.com/channel/UC2dG5T9SJkUFTy-h0KbVbkw