Prim1_secx
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The document derives an integral formula for sec3x in 3 steps. It first shows that the integral of secx dx equals ln|secx + tanx| + C. It then rewrites the integral of sec3x dx in terms of secx and tanx. Finally, it integrates by parts and combines the results with the first step to obtain the formula: ∫ sec3x dx = 1/2 ln|secx tanx| + tanx secx + C.
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Prim1_secx
- 1. Sabemos que: d tan x sec x+sec 2 x tan x +sec x ln∣sec x+tan x∣ = = sec x = sec x (1) dx sec x+tan x sec x +tan x isso implica ∫ sec x dx = ln∣sec x +tan x∣ + C (2) Agora podemos escrever ∫ sec3 x dx = ∫ sec2 x sec x dx = ∫ (1+tan2 x )sec x dx = ∫ sec x dx+∫ tan 2 x sec x dx (3) e integrando por partes obtemos ∫ sec3 x dx = ∫ sec2 x sec x dx = tan x sec x − ∫ tan2 x sec x dx (4) Somando (3) e (4) e aplicando (2), obtemos 1 1 ∫ sec3 x dx = 2 ln∣sec x tan x∣ + tan x sec x + C 2 (5)