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Math  /  Calculus

Question\int \frac{1}{x^2} \, dx

Studdy Solution

STEP 1

1. The integral is indefinite, meaning we are looking for the antiderivative.
2. The function 1x2\frac{1}{x^2} can be rewritten in a form that is easier to integrate.

STEP 2

1. Rewrite the integrand in a simpler form.
2. Integrate the function.
3. Add the constant of integration.

STEP 3

Rewrite the integrand 1x2\frac{1}{x^2} as a power of xx:
1x2=x−2\frac{1}{x^2} = x^{-2}

STEP 4

Integrate the function x−2x^{-2} using the power rule for integration, which states that ∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, where n≠−1n \neq -1:
∫x−2 dx=x−2+1−2+1+C=x−1−1+C\int x^{-2} \, dx = \frac{x^{-2+1}}{-2+1} + C = \frac{x^{-1}}{-1} + C

STEP 5

Simplify the expression:
x−1−1=−x−1=−1x\frac{x^{-1}}{-1} = -x^{-1} = -\frac{1}{x}

STEP 6

Add the constant of integration CC:
∫1x2 dx=−1x+C\int \frac{1}{x^2} \, dx = -\frac{1}{x} + C
The integral of 1x2\frac{1}{x^2} is:
−1x+C-\frac{1}{x} + C

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