Integrate x/(1+x): Step-by-Step Guide

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In summary, integration is the process of finding the antiderivative of a function, which is the inverse of differentiation. It is important because it allows us to solve various problems in fields like physics and economics, and it is a fundamental concept in calculus. To integrate x/(1+x), we can use the substitution method and the power rule to find the antiderivative, which is ln|1+x| + C. A step-by-step guide for integrating x/(1+x) would involve rewriting the integral, using the power rule, and substituting back the original variable. An example of integrating x/(1+x) would be finding the integral of this function from 0 to 1, which would result in ln(2) as
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jasmaster35
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How to integrate: x / (1+x)
 
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"By parts" would be my first stab at it
 
  • #3
Would integrating, [tex] \frac{1}{1+x} [/tex] be any easier?

If you got rid of the [itex] x [/itex] on the top it might make your life easier right?

[tex] \frac{x}{1+x} [/tex]

[tex] \frac{x}{1+x} - \frac{1+x}{1+x} = \frac{-1}{1+x} [/tex]

[tex] \frac{x}{1+x}=\frac{-1}{1+x}+\frac{1+x}{1+x} [/tex]

[tex] \frac{x}{1+x} = \frac{1+x}{1+x} - \frac{1}{1+x} [/tex]

[tex] \frac{x}{1+x} = 1 - \frac{1}{1+x} [/tex]

Obviously you don't have to do that in as many steps.
 
  • #4
jasmaster35 said:
How to integrate: x / (1+x)

Use the substitution 1 + x = t.
 

FAQ: Integrate x/(1+x): Step-by-Step Guide

1. What is integration?

Integration is a mathematical process that involves finding the antiderivative of a function. It is the inverse operation of differentiation and is used to calculate the area under a curve.

2. Why is it important to integrate?

Integration is important because it allows us to solve a wide variety of problems in fields such as physics, engineering, and economics. It is also a fundamental concept in calculus and helps us understand the relationship between a function and its rate of change.

3. How do you integrate x/(1+x)?

To integrate x/(1+x), we first use the substitution method by letting u = 1+x. Then, we can rewrite the integral as ∫(x/u) du. Using the power rule, we get the antiderivative as ln|u| + C. Finally, we substitute back u = 1+x to get the final answer of ln|1+x| + C.

4. What is the step-by-step guide for integrating x/(1+x)?

The following is a step-by-step guide for integrating x/(1+x):
1. Rewrite the integral as ∫(x/u) du, where u = 1+x
2. Use the power rule to find the antiderivative, which is ln|u| + C
3. Substitute back u = 1+x to get the final answer of ln|1+x| + C

5. Can you provide an example of integrating x/(1+x)?

Yes, for example, to integrate x/(1+x), we can follow the steps outlined in the previous question. The final answer would be ln|1+x| + C. For a specific example, let's say we want to find the integral of x/(1+x) from 0 to 1. Using the Fundamental Theorem of Calculus, we can evaluate the integral as ln|1+1| - ln|1+0| = ln(2) - ln(1) = ln(2).

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