How Do You Integrate 1/(x(ln x)^2) with Respect to x?

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In summary, integration for a linear function involves using the power rule, substitution is a common method for integration, there is a difference between definite and indefinite integration, integration can be used to find the area under a curve, and it can also be used to solve differential equations.
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sid_galt
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How to integrate this for x?

integral calc(1/x*1/(In x)^2*dx)
 
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Use the substitution: u=ln(x)
 
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To integrate this for x, we can use the substitution method. Let u = ln(x), then du = 1/x dx. Substituting this into the integral, we get:

∫ (1/x * 1/(ln x)^2 * dx) = ∫ (1/u^2 * du)

Using the power rule for integration, we get:

= -1/u + C = -1/ln(x) + C

Therefore, the final solution for integrating this for x is:

∫ (1/x * 1/(ln x)^2 * dx) = -1/ln(x) + C
 

FAQ: How Do You Integrate 1/(x(ln x)^2) with Respect to x?

1. How do I integrate this for a linear function?

For a linear function, integration involves using the power rule, where you increase the exponent by 1 and divide by the new exponent. For example, if the function is y = 3x, the integrated function would be y = 3x^2/2 + C, where C is the constant of integration.

2. Can I use substitution for integration?

Yes, substitution is a common method for integration. It involves replacing a variable in the function with a new variable that makes the integration process easier. This new variable is often denoted as u in the integration process.

3. What is the difference between definite and indefinite integration?

Indefinite integration is finding the antiderivative of a function, which means finding any function whose derivative is the given function. Definite integration, on the other hand, involves finding the exact numerical value of the integral within a specific interval.

4. How can I use integration to find the area under a curve?

To find the area under a curve, you can use the definite integral. The integral of a function over a given interval represents the area under the curve within that interval. So, by finding the definite integral over the desired interval, you can determine the area under the curve.

5. Can integration be used to solve differential equations?

Yes, integration can be used to solve differential equations. By integrating both sides of a differential equation, you can find a general solution that satisfies the equation. However, you may also need to use initial conditions to find a specific solution to the differential equation.

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