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jasmaster35
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How to integrate: x / (1+x)
jasmaster35 said:How to integrate: x / (1+x)
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.
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.
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.
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
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).