Integrate y= arctan x for 0< x <1

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In summary, the conversation discusses a question involving the integration of y = arctanx for 0 < x < 1 using the method of integration by parts and substitution. The individual was able to reach the point of x arctanx - integral of x/1+x^2 (between 0 and 1) but encountered difficulty in finding the correct answer. They tried using the substitution method of w = 1 + x^2 and dx = dw / 2x but did not get the right answer. However, it was pointed out that the substitution method should have worked and the individual later realized their mistake of writing w/2 instead of 1/2w.
  • #1
3ephemeralwnd
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the question: integrate y= arctanx for 0< x <1

i know to solve this question you should use the method of integral by parts, so i tried doing that and at some point i got
x arctanx - integral of x/1+x^2 (between 0 and 1)
.. from this point on i tried using the substitution method, w = 1 + x^2, and dx = dw / 2x
i plugged in those in and solved, but i didnt end up with the right answer

turns out the integral of x/1+x^2 is ln|1+x^2| ...
but shoudln't the method of substitution work as well? or did i do something wrong?
 
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  • #2
hi 3ephemeralwnd! :smile:

(try using the X2 icon just above the Reply box :wink:)
3ephemeralwnd said:
… x arctanx - integral of x/1+x^2 (between 0 and 1)
.. from this point on i tried using the substitution method, w = 1 + x^2, and dx = dw / 2x
i plugged in those in and solved, but i didnt end up with the right answer

should have worked :confused: …

(btw, it would be easier to write 2xdx = dw)

show us exactly what you did :smile:​
 
  • #3


oh sorry! i figured out my mistake

after the substituion i accidentally wrote it as w/2 instead of 1/2w

thanks for the reply though :)
 

1. What does the integral of y= arctan x for 0< x <1 represent?

The integral of y= arctan x for 0< x <1 represents the area under the curve of the function y= arctan x between the limits of 0 and 1 on the x-axis. This can also be interpreted as the total change in the value of y over this interval.

2. How do you solve the integral of y= arctan x for 0< x <1?

To solve this integral, you can use the integration by parts method or the substitution method. Both methods involve breaking down the integral into simpler parts and using mathematical rules and formulas to solve it.

3. What is the antiderivative of y= arctan x for 0< x <1?

The antiderivative (or primitive function) of y= arctan x for 0< x <1 is given by x*arctan(x)- ln(√(1+x^2)) + C, where C is a constant of integration. This can be verified by taking the derivative of the antiderivative and comparing it to the original function.

4. Why is the integral of y= arctan x for 0< x <1 important in calculus?

The integral of y= arctan x for 0< x <1 is important in calculus because it is a fundamental concept used in finding the area under curves and determining the total change in a function. It is also a key step in solving more complex integrals that cannot be solved using basic rules and formulas.

5. Can the integral of y= arctan x for 0< x <1 be approximated using numerical methods?

Yes, the integral of y= arctan x for 0< x <1 can be approximated using numerical methods such as the trapezoidal rule or Simpson's rule. These methods involve dividing the interval into smaller subintervals and using mathematical formulas to estimate the area under the curve.

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