{xe^x dx
integrate by parts let
f(x) = x so f'(x) = 1
and g(x) = e^x so g'(x) = e^x
so..
f(x)*g(x) - {(g(x)*f'(x)) dx
therefore xe^x - {(e^x * 1) dx
so..
xe^x - e^x + C
factorize so...
(x-1)e^x + C
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integral of e to the power -x is -e to the power -x
The power law of indices says: (x^a)^b = x^(ab) = x^(ba) = (x^b)^a → e^(2x) = (e^x)² but e^x = 2 → e^(2x) = (e^x)² = 2² = 4
d/dx (e-x) = -e-x
e^(-2x) * -2 The derivative of e^F(x) is e^F(x) times the derivative of F(x)
Integrate 2sin(x)cos(x)dxLet u = cos(x) and du = -sin(x)dx and pull out the -2:-2[Integral(u*du)]Integrate with respect to u:-2(u2)/2 + CSimplify:-u2 + CReplace u with cos(x):-cos2(x) + C