What is the integral of ln(x)
The integral of ln(x) dx is equal to x ln(x) - x + C, where C is the constant of integration.
To find the integral of ln(x), let's follow these steps:
Step 1: Start with the integral expression ∫ ln(x) dx.
Step 2: Rewrite ln(x) as ln(e^ln(x)), since ln(x) can be expressed as ln(e^x).
Step 3: Apply the properties of logarithms to simplify further: ln(e^ln(x)) = ln(x) * ln(e).
Step 4: Notice that ln(e) is equal to 1, so we can rewrite the expression as ∫ ln(x) * 1 dx.
Step 5: Now we can separate the variables by moving the constant 1 outside the integral: ∫ ln(x) dx = ∫ ln(x) * dx.
Step 6: Integrate the expression:
∫ ln(x) dx = x * ln(x) - ∫ x * (1/x) dx.
Step 7: Simplify further:
∫ ln(x) dx = x * ln(x) - ∫ dx.
Step 8: Evaluate the definite integral of dx, which equals x:
∫ ln(x) dx = x * ln(x) - x + C,
where C is the constant of integration.
Therefore, the integral of ln(x) is given by x * ln(x) - x + C.
To find the integral of ln(x), you can use integration by parts. The formula for integration by parts is:
∫u*dv = u*v - ∫v*du
Let's apply this formula to the integral of ln(x):
∫ln(x) dx
Step 1: Choose u and dv
Let u = ln(x) and dv = dx
Step 2: Find du and v
To find du, differentiate u with respect to x:
du = (1/x) dx
To find v, integrate dv with respect to x:
v = ∫dx = x
Step 3: Apply the formula
Using the integration by parts formula: ∫u*dv = u*v - ∫v*du
∫ln(x) dx = ln(x) * x - ∫x * (1/x) dx
Simplifying further, we get:
∫ln(x) dx = x * ln(x) - ∫dx
The integral of dx is simply x. So, our final result is:
∫ln(x) dx = x * ln(x) - x + C
where C is the constant of integration.