∫ (1)/(x²+1) dx
To find the integral of 1/(x^2+1) dx, we can use a trigonometric substitution
To find the integral of 1/(x^2+1) dx, we can use a trigonometric substitution.
Let’s make the substitution x = tan(theta). In this case, dx = sec^2(theta) d(theta), and x^2+1 = tan^2(theta)+1 = sec^2(theta).
Substituting these values into the integral, we get:
∫ (1)/(x^2+1) dx = ∫ (1)/(sec^2(theta)) * sec^2(theta) d(theta)
= ∫ d(theta)
= theta + C
Now, we need to find theta. From the substitution we made (x = tan(theta)), we can rearrange it to get theta = arctan(x).
Therefore, the final answer is ∫ (1)/(x^2+1) dx = arctan(x) + C, where C is the constant of integration.
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