![]() Each integral will be dealt with differently. Strip 2 secants out and convert rest to tangents using 2 2 sec 1 tan x x = +, then use the substitution tan u x =. Strip 1 tangent and 1 secant out and convert the rest to secants using 2 2 tan sec 1 x x = -, then use the substitution sec u x =. For tan sec n m x x dx we have the following : 1. Use double angle and/or half angle formulas to reduce the integral into a form that can be integrated. Strip 1 cosine out and convert rest to sines using 2 2 cos 1 sin x x = -, then use the substitution sin u x =. Strip 1 sine out and convert rest to cosines using 2 2 sin 1 cos x x = -, then use the substitution cos u x =. 5 3 ln x dx 1 ln x u x dv dx du dx v x = ( ) ( ) () () 5 5 5 5 3 3 3 3 ln ln ln 5ln 5 3ln 3 2 x dx x x dx x x x = - = - = - Products and (some) Quotients of Trig Functions For sin cos n m x x dx we have the following : 1. x x dx - e x x u x dv du dx v - =- e e x x x x x x dx x dx x c - =- + =- + e e e e e Ex. Choose u and dv from integral and compute du by differentiating u and compute v using v dv =. ( ) 2 3 2 1 5 cos x x dx 3 2 2 1 3 3 u x du x dx x dx du = 3 3 1 1 1 :: 2 2 8 x u x u = ( ) ( ) ( ) () () ( ) 2 3 2 8 5 3 1 1 8 5 5 3 3 1 5 cos cos sin sin 8 sin 1 x x dx u du u = - Integration by Parts : u dv uv v du = - and b b b a a a u dv uv v du =. For indefinite integrals drop the limits of integration. u Substitution : The substitution ( ) u gx = will convert ( ) ( ) ( ) ( ) ( ) ( ) b gb a ga f g x g x dx f u du ¢ = using ( ) du g x dx ¢ =. © 2005 Paul Dawkins Standard Integration Techniques Note that at many schools all but the Substitution Rule tend to be taught in a Calculus II class. If ( ) ( ) f x gx ‡ on a x b £ £ then ( ) ( ) b b a a f x dx g x dx ‡ If ( ) 0 f x ‡ on a x b £ £ then ( ) 0 b a f x dx ‡ If ( ) m f x M £ £ on a x b £ £ then ( ) ( ) ( ) b a mb a f x dx M b a - £ £ - Common Integrals k dx kx c = + 1 1 1, 1 n n n x dx x cn + + = + „- 1 1 ln x x dx dx x c - = + 1 1 ln a ax b dx ax b c + = + + ( ) ln ln u du u u u c = - + u u du c = + e e cos sin u du u c = + sin cos u du u c =- + 2 sec tan u du u c = + sec tan sec u u du u c = + csc cot csc u udu u c =- + 2 csc cot u du u c =- + tan ln sec u du u c = + sec ln sec tan u du u u c = + + ( ) 1 1 1 2 2 tan u a a a u du c - + = + ( ) 1 2 2 1 sin u a a u du c - = + Calculus Cheat Sheet Visit for a complete set of Calculus notes. Variants of Part I : () ( ) ( ) ( ) ux a d f t dt u x f ux dx ¢ = Ø ø º ß () ( ) ( ) ( ) b vx d f t dt v x f vx dx ¢ =- Ø ø º ß () ( ) ( ) ( ) ( ) () () ux vx ux vx d f t dt u x f v x f dx ¢ ¢ = - Properties ( ) ( ) ( ) ( ) f x g x dx f x dx g x dx – = – ( ) ( ) ( ) ( ) b b b a a a f x g x dx f x dx g x dx – = – ( ) 0 a a f x dx = ( ) ( ) b a a b f x dx f x dx =- ( ) ( ) cf x dx c f x dx =, c is a constant ( ) ( ) b b a a cf x dx c f x dx =, c is a constant ( ) b a c dx cb a = - ( ) ( ) b b a a f x dx f x dx £ ( ) ( ) ( ) b c b a a c f x dx f x dx f x dx = + for any value of c. ( ) ( ) F x f x dx = ) then ( ) () ( ) b a f x dx Fb Fa =. Part II : ( ) f x is continuous on, ab, ( ) F x is an anti-derivative of ( ) f x ( i.e. Fundamental Theorem of Calculus Part I : If ( ) f x is continuous on, ab then ( ) () x a gx f t dt = is also continuous on, ab and ( ) () ( ) x a d g x f t dt f x dx ¢ =. ![]() Indefinite Integral : ( ) ( ) f x dx F x c = + where ( ) F x is an anti-derivative of ( ) f x. Anti-Derivative : An anti-derivative of ( ) f x is a function, ( ) F x, such that ( ) ( ) F x f x ¢ =. ![]() Then ( ) ( ) * 1 lim i b a n i f x dx f x x fi¥ = ¥ = D. Divide, ab into n subintervals of width x D and choose * i x from each interval. © 2005 Paul Dawkins Integrals Definitions Definite Integral: Suppose ( ) f x is continuous on, ab. Calculus Cheat Sheet Visit for a complete set of Calculus notes.
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