what is integration of e^x/1+e^2x
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solve the integral using substitution technique.
Let e^x = t => e^x dx = dt
e^x dx/(e^2x +1) = dt/(t^2 + 1) = arctan t + C
e^x dx/(e^2x +1) = arctan e^x + C
.......... OR
First, recall that the derivative of arctan(u) is u' / (1+u^2).
Do you see the similarity?
Let u = e^x, then du = e^x dx.
e^(2x) = (e^x)^2 = u^2
The integral becomes
du / (1+u^2)
= arctan(u) + C
= arctan(e^x) + C
Hope it helps
Let e^x = t => e^x dx = dt
e^x dx/(e^2x +1) = dt/(t^2 + 1) = arctan t + C
e^x dx/(e^2x +1) = arctan e^x + C
.......... OR
First, recall that the derivative of arctan(u) is u' / (1+u^2).
Do you see the similarity?
Let u = e^x, then du = e^x dx.
e^(2x) = (e^x)^2 = u^2
The integral becomes
du / (1+u^2)
= arctan(u) + C
= arctan(e^x) + C
Hope it helps
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