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MathematicalPhysicist
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what is the integral of lnx?
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Integrating Natural Logarithms: A Scientific Approach
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In summary, the integral of ln(x) is found by writing it as 1*ln(x), then integrating by parts. The final result is x*ln(x) - x (+ C).
- #2
Lonewolf
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Try writing it as 1*ln(x), then integrate by parts. (Should get x*ln(x)-x).
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MathematicalPhysicist
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could you show me the entire way?
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Lonewolf
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Sure
∫1*ln(x)dx
du/dx = 1
v = ln(x)
u = x
dv/dx = 1/x
Now, integrating by parts:
∫1*ln(x)dx = u*v - ∫u*dv/dx
= x*ln(x) - ∫dx
= x*ln(x) - x (+ C to be pedantic)
∫1*ln(x)dx
du/dx = 1
v = ln(x)
u = x
dv/dx = 1/x
Now, integrating by parts:
∫1*ln(x)dx = u*v - ∫u*dv/dx
= x*ln(x) - ∫dx
= x*ln(x) - x (+ C to be pedantic)
- #5
MathematicalPhysicist
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i got it, thanks.
edit: just one problem should this ∫u*dv/dx=∫1
edit: just one problem should this ∫u*dv/dx=∫1
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MathematicalPhysicist
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lonewolf i think i am wrong but i need your verification on that.
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MathematicalPhysicist
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never mind i found my mistake.
- #8
Lonewolf
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Oops, sorry LQG, I wasn't ignoring you intentionally. Glad you figured it out anyway 
1. What is the meaning of the integral of lnx?
The integral of lnx, also known as the natural logarithmic function, is the inverse of the derivative of lnx. It represents the area under the curve of the function y=lnx from a given interval.
2. How is the integral of lnx calculated?
The integral of lnx can be calculated using the integration technique known as integration by parts, where u=lnx and dv=dx. The formula for integration by parts is ∫udv = uv - ∫vdu. By plugging in u=lnx and dv=dx, the integral can be solved step by step.
3. What is the importance of the integral of lnx in mathematics?
The integral of lnx is important in mathematics as it has many applications in various fields such as physics, economics, and engineering. It is used to solve problems involving rates of change, growth and decay, and optimization.
4. Can the integral of lnx be solved without using integration by parts?
Yes, the integral of lnx can also be solved using substitution or other integration techniques such as u-substitution. However, integration by parts is the most commonly used method for solving this integral.
5. Is there a specific limit to the interval when solving the integral of lnx?
Yes, the interval for solving the integral of lnx is typically from 0 to infinity. This is because the natural logarithmic function is undefined for negative numbers and the integral is only valid for positive intervals.
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