Integration by Parts

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The general formula to do integration by parts:

∫u*dv=uv-∫v*du or ∫f(x)g'(x)dx=f(x)g(x)-∫g(x)f'(x)dx


EXAMPLE 1: Find the integral of xsin(x)

Step 1. Define f(x), g'(x), f'(x), g(x)

f(x)=x

g'(x)=sin(x)

f'(x)=1

g(x)=-cos(x)

Then using the formula, we can integrate.

∫xsin(x)dx=-xcos(x)-∫-cos(x)dx

= -xcos(x)+∫ cos(x)dx = -xcos(x)+ sin(x)+C **Don't forget the constant!**


EXAMPLE 2:

Find the integral of ∫x2ln(x)dx

Step 1. Define f(x), g'(x), f'(x), g(x)

f(x)=ln(x)

g'(x)=x2

f'(x)=1/x

g(x)=1/3 x3

∫ln(x)*x2 dx= 1/3 ln(x)x3-∫ 1/x *1/3 x3 dx

Then using the formula, we can integrate.

= 1/3 ln(x)x3 - 1/12 ln(x) x4+C ***Don't forget Constant***