Given: f(x) = x + 2/x - 3/x^2f(x)=x+2x−3x2
Required: Antiderivative of f(x)f(x)
Theorems, Definitions and Principles:
A function F(x)F(x) is an antiderivative or an indefinite integral of the function f(x)f(x) if the derivative F' = f. We use the notation
color(brown)(F(x)=intf(x)dx)
to indicate that F is an indefinite integral of f. Using this notation, we have color(brown)(F(x)=intf(x)dx) if and only if color(brown)(F'(x)=f(x)) or color(brown)(f(x) = d/(dx)[intf(x)dx])
Solution Strategy: Apply the above theorem
f(x) = x + 2/x - 3/x^2
thus the antiderivative F(x)= intf(x)=int(x + 2/x - 3/x^2) dx
Now to integrate apply linearity and power rule and knowledge of standard integrals:
power rule => intx^ndx = 1/(n+1)x^(n+1)
F(x)= intx dx + int2/x dx - int3/x^2 dx Apply linearity
F_1(x) = intx dx= x^2/2+C_1 " :" Apply power rule n=2
F_2(x) = 2int1/x dx= 2lnx +C_2" :" linearity and standard integral
F_1(x)=3int1/x^2 dx= -3/x+C_3" :"Apply power rule n=-2
Regroup all together and the antiderivative is:
F(x) =x^2/2 + 2lnx + 3/x+C
C consolidates the other constants C_1, C_2, C_3
Check: (dF(x))/(dx)= cancel2x/cancel2+2/x-3/(x^2)=x+2/x+3/x^2
