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Integration

⌠          1 
⎮xⁿ.dx = ───── xⁿ⁺¹ + C
⌡        (n+1)


⌠ 1 
⎮───.dx = logx + C
⌡ x



⌠
⎮eˣ.dx = eˣ + C
⌡


⌠                   ⌠          ⌠               
⎮[f(x) ± g(x)].dx = ⎮f(x).dx ± ⎮g(x).dx     
⌡                   ⌡          ⌡               


⌠
⎮sinx.dx = -cosx + C
⌡


⌠
⎮cosx.dx = sinx + C
⌡

Integration by parts

⌠           ⌠       ⌠   ⎛⌠    ⎞     
⎮u v dx =  u⎮v dx - ⎮u' ⎜⎮v.dx⎟ . dx
⌡           ⌡       ⌡   ⎝⌡    ⎠     

where u' = du/dx

To choose u (first function) and v (second function), use the 'ILATE' trick: -I: Integral -L: Logarithm -A: Algebraic -T: Trigonometric -E: Exponent

Example:

   ⌠          
   ⎮x.cosx.dx 
   ⌡          

Algebraic and trigonometric parts. ILATE => A comes first.
ie,
 - u = x
 - v = cosx

    ⌠          ⌠dx ⎛⌠       ⎞ 
=  x⎮cosx.dx - ⎮── ⎜⎮cosx.dx⎟ . dx
    ⌡          ⌡dx ⎝⌡       ⎠


            ⌠ ⎛⌠       ⎞       
=  x.sinx - ⎮ ⎜⎮cosx.dx⎟ . dx
            ⌡ ⎝⌡       ⎠


            ⌠       
=  x.sinx - ⎮sinx.dx
            ⌡       


=  x.sinx + cosx

Derivatives

Product rule

Derivative of:

f(x).g(x)

with respect to x is:

g.f'(x) + f.g(x)

Or in Leibniz's notation:

 d  ⎛   ⎞      df       dg
─── ⎜f.g⎟ = g. ─── + f. ───
 dx ⎝   ⎠      dx       dx

Chain rule

Derivative of f(g(x)) with respect to x is:

f'(g(x)) . g'(x)