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hi i need help to integrate the following:?
e^x/(1+e^2x) dx
(1/x^2)(sec(1/x))^2 dx
2 Answers
- Anonymous1 decade ago
i) ∫e^x dx / (1 + e^(2x))
let u = e^x, du = e^x dx
∫(du / (1 + u^2)) = arctan(u) + C = arctan(e^x) + C
ii) ∫(sec^2(1/x) dx / x^2)
let u = (1/x), du = -dx/x^2
= - ∫sec^2(u) du
= - tan(u) + C
= - tan(1/x) + C.
- FazLv 71 decade ago
e^x / (1+e^(2x)) dx
Let u=e^x, then du=e^x dx, so dx = 1/e^x du = (1/u)du
= ∫ u / (1+ u²) (1/u) du
= ∫ 1 / (1+u²) du
= tan‾ ¹(u) + c
= tan‾ ¹( e^x ) + c
∫ (1/x²) sec²(1/x)) dx
Let u=1/x then du = 1/x² dx, so dx = x² du
= ∫ - (1/x²) sec²(u) x² du
= ∫ sec²(u) du
= -tan(u) + c
= -tan(1/x) + c
