What is to integrate a function? Integrate the function y=(2x^3+3x+1)/x^4

To integrate a function f(x) means to determine the
function F(x), which, when is differentiated, gives back the function
f(x).

We'll integrate f(x) =
y.

Int f(x)dx  =Int (2x^3+3x+1)
dx/x^4

We'll apply the linear property of integrals and
we'll get:

Int (2x^3+3x+1) dx/x^4 = Int 2x^3dx/x^4 + Int
3xdx/x^4 + Int dx/x^4

We'll calculate each
Integral:

We'll start with Int
2x^3dx/x^4

We'll reduce like
terms:

Int 2x^3dx/x^4 = Int
2dx/x

Int 2dx/x = 2Int
dx/x

2Int dx/x = 2ln x +
C

Now, we'll calculate Int
3xdx/x^4.

Int 3xdx/x^4 = Int
3dx/x^3

Int 3dx/x^3 = 3Int
x^-3*dx

3Int x^-3*dx = 3*x^(-3+1)/(-3+1) +
C

3*x^(-3+1)/(-3+1) + C = -3/2x^2 +
C

Finally, we'll calculate Int
dx/x^4:

Int dx/x^4 = Int
x^-4*dx

Int x^-4*dx = x^(-4+1)/(-4+1) +
C

x^(-4+1)/(-4+1) + C = -1/3x^3 +
C

Now, the Integrals is:

Int
f(x)dx = 2ln x - 3/2x^2 - 1/3x^3 + C

Int
f(x)dx = ln x^2 - 3/2x^2 - 1/3x^3 + C