Solving the Integral of 1/x: Using the Natural Logarithm Function

∫ 1/x dx

To solve the integral of 1/x dx, we can use the natural logarithm function

To solve the integral of 1/x dx, we can use the natural logarithm function.

The integral of 1/x can be denoted as ∫ 1/x dx, which is equivalent to the integral of x^(-1) dx. Using the power rule for integration, where ∫ x^n dx = (x^(n+1))/(n+1) + C, where C is the constant of integration.

In this case, when n = -1, we get:

∫ 1/x dx = (x^(-1+1))/(-1+1) + C
= (x^0)/0 + C
= 1/0 + C

However, 1/0 is undefined because division by zero is not defined in mathematics. Therefore, the integral of 1/x is a special case and cannot be easily solved using the power rule.

Instead, we can use the property of the natural logarithm to find a solution. The natural logarithm function is defined as ln(x), where ln(x) represents the logarithm base e (with e being the Euler’s number) of x.

Using this property, we can rewrite the integral as:

∫ 1/x dx = ∫ x^(-1) dx
= ln|x| + C

Here, the absolute value is used to handle negative values of x. The constant of integration, C, represents any arbitrary constant.

Therefore, the integral of 1/x is ln|x| + C, where ln|x| represents the natural logarithm of the absolute value of x.

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