Integral of 2/(Y+1) - Solution without Calculator

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SUMMARY

The integral of 2/(Y+1) over the interval [2,0] can be solved analytically by finding the antiderivative, which is ln(Y+1). The correct evaluation of the integral results in 2ln(3). The process involves substituting the limits of integration into the antiderivative and calculating the difference between the function values at these endpoints. This method eliminates the need for a calculator and relies on fundamental integration techniques.

PREREQUISITES
  • Understanding of basic calculus concepts, particularly integration.
  • Familiarity with antiderivatives, specifically ln(x) and ln(x+1).
  • Knowledge of the substitution method in integration.
  • Ability to evaluate definite integrals and apply limits of integration.
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  • Study the properties of logarithmic functions and their derivatives.
  • Learn about the substitution method in integral calculus.
  • Practice evaluating definite integrals with various functions.
  • Explore advanced integration techniques, such as integration by parts and partial fractions.
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Students and educators in calculus, mathematicians seeking to enhance their integration skills, and anyone interested in solving integrals without computational tools.

Whitebread
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Hello,
I need to take the integral of 2/(Y+1) over the interval [2,0]. The only analytical method I know is to take the anti-deriviative of the integrand and subtract the function values at the endpoints of the interval. I know that the anti-deriviate of (x)^-1 is ln(x), but I don't know how to do this problem without using the calculator (and I'd rather not do that). Can someone help?
 
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Whitebread said:
Hello,
I need to take the integral of 2/(Y+1) over the interval [2,0]. The only analytical method I know is to take the anti-deriviative of the integrand and subtract the function values at the endpoints of the interval. I know that the anti-deriviate of (x)^-1 is ln(x), but I don't know how to do this problem without using the calculator (and I'd rather not do that). Can someone help?

well y' = 2/(Y+1) remember that

Integrate:
2/(Y+1) * y'/y'
2/y' * ln(y+1)
substitute y'
(y+1)ln(y+1)|2,0
[3ln3 - 0]
3ln3 is the answer
hope i did it right...
 
Whitebread said:
Hello,
I need to take the integral of 2/(Y+1) over the interval [2,0]. The only analytical method I know is to take the anti-deriviative of the integrand and subtract the function values at the endpoints of the interval. I know that the anti-deriviate of (x)^-1 is ln(x), but I don't know how to do this problem without using the calculator (and I'd rather not do that). Can someone help?


well, clearly if you know that the antiderivative of 1/x is ln (x) then the antiderivative of 1/(x+1) is ln (1 +x) (check! Take the derivative of the result!). and the antiderivative of 1/(Ax+B) with A and B being constants is ln(A +B x) /A (check!).

The way to prove it is to simply do the obvious change of variable, u = 1+x, dx=du and then integrate over u, which you know how to do. (If you do this change of variable to do yoru problem, don't forget to change the limits of integration.)
 
Last edited:
Remember that dx=d(x+c), c=const.

- Kamataat
 
Hi guys,
the correct answer is 2ln3, thanks for the help guys!
 

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