Cheat_Sheet

15.17.19.21.23.25.26.29.33.36.∫∫∫∫∫∫∫∫∫∫Theoretical Computer Science Cheat SheetCalculus Cont.arccos x a dx = arccos x a − √ ∫a 2 − x 2 , a > 0, 16. arctan x a dx = x arctan x a − a 2 ln(a2 + x 2 ), a > 0,∫sin 2 ( )(ax)dx = 12a ax − sin(ax) cos(ax) , 18. cos 2 )(ax)dx =2a( 1 ax + sin(ax) cos(ax) ,∫sec 2 xdx= tan x, 20. csc 2 xdx= − cot x,sin n xdx= − sinn−1 x cos x+ n − 1 ∫∫sin n−2 x dx, 22. cos n xdx= cosn−1 x sin x+ n − 1 ∫cos n−2 x dx,n nn n∫∫∫tan n xdx= tann−1 x− tan n−2 x dx, n ≠1, 24. cot n xdx= − cotn−1 x− cot n−2 x dx, n ≠1,n − 1n − 1sec n xdx= tan x secn−1 xn − 1+ n − 2 ∫n − 1sec n−2 x dx, n ≠1,csc n xdx= − cot x cscn−1 x+ n − 2 ∫∫∫csc n−2 x dx, n ≠1, 27. sinh xdx= cosh x, 28. cosh xdx= sinh x,n − 1 n − 1∫∫∫tanh xdx=ln| cosh x|, 30. coth xdx=ln| sinh x|, 31. sech xdx= arctan sinh x, 32. csch xdx=ln ∣ tanhx∣ ,sinh 2 xdx= 1 4 sinh(2x) − 1 2 x,34. ∫cosh 2 xdx= 1 4 sinh(2x)+ 1 2 x,arcsinh x a dx = x arcsinh x a − √ x 2 + a 2 , a > 0, 37.∫35. ∫sech 2 xdx= tanh x,arctanh x a dx = x arctanh x a + a 2 ln |a2 − x 2 |,238.39.40.42.43.46.48.50.52.54.56.58.60.⎧∫⎨ x arccosh xarccosh x a dx = a − √ x 2 + a 2 ,⎩x arccosh x a + √ x 2 + a 2 ,∫dx(√a2 + x =ln x + √ )a 2 + x 2 , a > 0,2∫∫if arccosh x a> 0 and a>0,if arccosh x a< 0 and a>0,∫dx√a2a 2 + x 2 = 1 a arctan x a , a > 0, 41. √− x 2 dx = x 2 a2 − x 2 + a22 arcsin x a , a > 0,(a 2 − x 2 ) 3/2 dx = x 8 (5a2 − 2x 2 ) √ a 2 − x 2 + 3a48 arcsin x a , a > 0,∫∫dx√a2 − x = arcsin x 2 a , a > 0, 44. dxa 2 − x 2 = 1 ∣ ∣∣∣2a ln a + xa − x∣ ,∫ √a2 √ ∣± x 2 dx = x ∣∣x2 a2 ± x 2 ± a22 ln √ ∣ ∫∣∣+ a2 ± x 2 , 47.∫dxax 2 + bx = 1 ∣ ∣∣∣a ln xa + bx∣ ,∫45. dx(a 2 − x 2 ) = x3/2 a 2√ a 2 − x , 2dx∣ ∣∣x√x2 − a =ln √ ∣ ∣∣ + x2 − a 2 , a > 0,2∫49. x √ 2(3bx − 2a)(a + bx)3/2a + bx dx =15b 2 ,∫ √a + bxdx =2 √ ∫∫1a + bx + axx √ a + bx dx,51. x√ dx = 1√ √ ∣ √ lna + bx − a∣∣∣a + bx 2∣√ √ , a > 0,a + bx + a∫ √a2 − x 2dx = √ ax2 − x 2 a + √ a− a ln2 − x 2∫∣ x ∣ ,53. x √ a 2 − x 2 dx = − 1 3 (a2 − x 2 ) 3/2 ,∫x 2√ a 2 − x 2 dx = x 8 (2x2 − a 2 ) √ ∫∣a 2 − x 2 + a48 arcsin x a , a > 0, 55. dx∣∣∣∣√a2 − x = − 1 2 a ln a + √ a 2 − x 2x ∣ ,∫∫xdx√a2 − x = −√ a 2 − x 2 x 2 dx, 57. √ 2 a2 − x = − √ x2 2 a2 − x 2 + a22 arcsin x a,a>0,∫ √a2 + x 2dx = √ ax2 + x 2 a + √ a− a ln2 + x 2∫ √ ∣ x ∣ , 59. x2 − a 2dx = √ xx2 − a 2 − a arccos a|x| , a > 0,∫x √ ∫∣ ∣x 2 ± a 2 dx = 1 3 (x2 ± a 2 ) 3/2 dx∣∣∣, 61.x √ x 2 + a = 1 2 a ln x ∣∣∣a + √ ,a 2 + x 2