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Strip 1 cosine out and convert rest to sines using 2 2cos 1 sin x x , then use the substitution sin u x. Strip 1 sine out and convert rest toĬosines using 2 2sin 1 cos x x , then use the substitution cos u x.

Products and (some) Quotients of Trig Functions For sin cos n m Integral and compute du by differentiating u and compute v using v dv .

Integration by Parts : u dv uv v du   andĪa a u dv uv v du  . For indefinite integrals drop the limits of integration. U Substitution : The substitution   u g x will convert            b g b Standard Integration Techniques Note that at many schools all but the Substitution Rule tend to be taught in a Calculus II class. Sin cos u du u c   2sec tan u du u c Ĭsc cot csc u udu u c   2csc cot u du u c   If   m f x M  on a x b  then       b If     f x g x on a x b  then     b b     cf x dx c f x dx , c is a constantĪ a cf x dx c f x dx , c is a constantĪ a c f x dx f x dx f x dx    for any value of c. Properties         f x g x dx f x dx g x dx     Part II :   f x is continuous on , a b,   F x isĪn anti-derivative of   f x ( i.e. Part I : If   f x is continuous on  , a b thenĪ g x f t dt  is also continuous on  , a b Where   F x is an anti-derivative of   f x. Indefinite Integral :     f x dx F x c 

Divide  , a b into n subintervals of width x and choose *Īnti-Derivative : An anti-derivative of   f x is a function,   F x, such that     F x f x . © 2005 Paul Dawkinsĭefinite Integral: Suppose   f x is continuous on  , a b. Visit for a complete set of Calculus notes.