Questions: ∫ 1/x dx = lnx + C

Transcript text: \[ \int \frac{1}{x} d x=[?]|x|+C \]

Solution

Solution Steps

Step 1: Identify the Integral

The given integral is:

\[ \int \frac{1}{x} \, dx \]

Step 2: Recall the Antiderivative Formula

The antiderivative of \(\frac{1}{x}\) is a well-known result. The integral of \(\frac{1}{x}\) with respect to \(x\) is the natural logarithm of the absolute value of \(x\):

\[ \int \frac{1}{x} \, dx = \ln |x| + C \]

where \(C\) is the constant of integration.

Step 3: Compare with the Given Expression

The problem provides the expression:

\[ \int \frac{1}{x} \, dx = [?]|x| + C \]

By comparing this with the known result, we can see that the missing part \([?]\) should be \(\ln\).

Final Answer

\[ \boxed{\ln |x| + C} \]

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