U subtitution for Arc Tan(x)/ x^2 +1 dx
To solve the integral ∫ arctan(x) / (x^2 + 1) dx using the method of u-substitution, follow these steps:
Step 1: Identify the function within the integral that can be the derivative of another function (u).
In this case, the function arctan(x) has a derivative of 1 / (x^2 + 1) with respect to x. Therefore, we will let u = arctan(x).
Step 2: Compute the derivative of u with respect to x.
Since u = arctan(x), du/dx = 1 / (x^2 + 1).
Step 3: Rearrange the equation to solve for dx.
Rearranging du/dx = 1 / (x^2 + 1), we get dx = 1 / (1 + x^2) du.
Step 4: Substitute u and dx in terms of du in the integral.
The integral becomes ∫ (u / (1 + x^2)) dx = ∫ u / (1 + x^2) * (1 / (1 + x^2)) du = ∫ u / (1 + x^2)^2 du.
Step 5: Evaluate the integral with respect to u.
Since we have transformed the integral with respect to x into an integral with respect to u, we can now solve it as a regular integral. So, the final integral is ∫ u / (1 + x^2)^2 du.
By following these steps, we have successfully transformed the original integral to a new form that can be more easily evaluated.