Given y'=e^(arctan x)/(x^2+1), what is the function y?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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To determine the primitive function, we'll have to calculate the indefinite integral of y'.

We'll apply substitution technique, replacing arctan x by t.

arctan x = t

We'll differentiate both sides and we'll get:

dx/(1 + x^2) = dt

We'll re-write the integral in the new variable:

Int e^(arctan x) dx/(1 + x^2) =Int e^t*dt

Int e^t*dt = e^t + C

Int e^(arctan x) dx/(1 + x^2) = e^(arctan x) + C

The primitive function is: y = e^(arctan x) + C