Find the indefinite integral of `g(x) = (19 + 43x^3)/x` , (x>0)

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Evaluate the indefinite integral of the function `g(x) = 19 + 43x^3/x, ` yields:

`int g(x) dx = int (19 + 43x^3/x)dx`

`int g(x) dx = int (19 + 43x^2)dx`

Using the property of linearity of integral, you need to split the integral into two easier integrals, such that:

`int g(x) dx = int (19) dx + int (43x^2)dx`

`int g(x) dx =19 x + 43x^3/3 + c`

Evaluate the indefinite integral of the function `g(x) = (19 + 43x^3)/x` , yields:

`int g(x) dx = int (19 + 43x^3)/x dx`

Using the property of linearity of integral, you need to split the integral into two easier integrals, such that:

`int g(x) dx = int (19/x) dx + int (43x^3)/x dx`

`int g(x) dx = 19 ln|x| + 43x^3/3 + c`

**Hence, evaluating the indefinite integral of the function `g(x) = 19 + 43x^3/x` yields int `g(x) dx =19 x + 43x^3/3 + c` and evaluating the indefinite integral of the function `g(x) = (19 + 43x^3)/x` yields int `g(x) dx = 19 ln|x| + 43x^3/3 + c.` **

The integral `int g(x) dx` has to be determined given that `g(x) = (19 + 43x^3)/x`

`int g(x) dx`

=> `int (19 + 43x^3)/x dx`

=> `int 19/x + 43x^2 dx `

=> `int 19/x dx + int 43x^2 dx`

=> `19*ln (x) + (43/3)*x^3 + C`

As it is given than x > 0, ln x is defined for all values of x

**The integral **`int (19 + 43x^3)/x dx =19*ln (x) + (43/3)*x^3 + C`

`intg(x)dx=int(19+43x^3)/xdx`

`=int(19/x+(43x^3)/x)dx`

`=int19/xdx+int(43x^2)dx`

`=19int(1/x)dx+43intx^2dx`

`=19ln(x)+43x^(2+1)/(2+1)+c`

`=19ln(x)+(43/3)x^3+c`

ans.

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