Integral Math Example 4

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Example 4

hard
Find ∫1x dx\int \frac{1}{x} \, dx and explain why the standard power rule does not apply.

Solution

  1. 1
    The power rule ∫xn dx=xn+1n+1+C\int x^n\,dx = \frac{x^{n+1}}{n+1}+C fails when n=−1n=-1 because it would give division by zero.
  2. 2
    The antiderivative of 1x\frac{1}{x} is the natural logarithm: ∫1x dx=ln⁡∣x∣+C\int \frac{1}{x}\,dx = \ln|x| + C.

Answer

ln⁡∣x∣+C\ln|x| + C
The power rule breaks down at n=−1n=-1 because dividing by n+1=0n+1=0 is undefined. The natural logarithm fills this gap: its derivative is 1x\frac{1}{x}, making it the correct antiderivative. The absolute value handles negative xx.

About Integral

The reverse operation of differentiation; it also computes the exact area under a curve between two points.

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