Understanding the Integral of 1/x | A Fundamental Logarithmic Function

∫ 1/x dx

The integral of 1/x with respect to x is a fundamental logarithmic function:

∫ 1/x dx = ln|x| + C

Here, ln|x| represents the natural logarithm of the absolute value of x, and C is the constant of integration

The integral of 1/x with respect to x is a fundamental logarithmic function:

∫ 1/x dx = ln|x| + C

Here, ln|x| represents the natural logarithm of the absolute value of x, and C is the constant of integration.

To understand why the integral of 1/x is ln|x|, we can use the concept of the inverse function of differentiation. Let’s differentiate ln|x+1| and see what we get:

d/dx ln|x| = 1/x

By taking the integral of 1/x, we are essentially asking what function differentiates to give us 1/x. And it turns out that ln|x| satisfies this requirement.

The absolute value of x is used in the natural logarithm to ensure that the function remains defined for both positive and negative values of x. Remember that the derivative of ln|x| is still 1/x.

The constant of integration, C, is added because when we differentiate the function ln|x|, the constant disappears. Therefore, when we integrate 1/x, we need to include the constant back.

To illustrate, let’s calculate the integral of 1/x:

∫ 1/x dx = ln|x| + C

So, the integral of 1/x with respect to x is ln|x| + C.

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