Jacob Barondes INTEGRATION BY PARTS: INDEFINITE

DEFINITION Indefinite Integration is the practice of performing indefinite integration ∫u dv This is done by expanding the equation into parts, uv and v du At “+c” at the end of each solution because what was found is just an approximation, nothing was actually evaluated. c is a constant

∫udv= uv- ∫vdu U is chosen such that du/dx is simpler u Du is the derivative of u Dv is the anti derivative of v EQUATION

SELECTING U AND DV Choose u in this general order 1.Logarithmic function 2. Inverse trig function 3.Algebraic function 4.Trig function 5.Exponential function

STEPS Choose u and v Differentiate u: du Integrate v: ∫v dx Put u, du and ∫v dx here: u∫v dx −∫u' (∫v dx) dx Simplify and solve

EX1. ∫xsinx dx U=x Du=dx Dv=sinx dx V= -cosx = -xcosx- ∫-cosx dx = -xcosx- (-sinx) +c =-xcosx+sinx+c

INTEGRATION BY PARTS TWICE Not all integration by parts problems can be solved in “integration” If doing the integration by parts leads to a ∫vdu as complicated as ∫2te t dt, that is a another integration by parts problem in itself.

EX 2. ∫x 2 e x dx= uv-∫vdu U=x 2 du=2xdx Dv=e x dx v=e x =x 2 e x - ∫e x 2xdx =x 2 e x - 2∫e x xdx ∫xe x dx = xe x -∫e x dx =xe x -e x -c Now we put it all together…. =x 2 e x -2(xe x -e x -c) =x 2 e x -2xe x -2e x +2c =x 2 e x -2xe x -2e x +c Where c is a constant.

PRACTICE PROBLEMS 1. ∫lnx/(x 2 ) dx 2. ∫arcsin3x dx

ANSWERS 1. –lnx/x-1/x+c 2. xarcsin3x+1/3√1-9x 2 +c

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