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spyros
 2 years ago
integral(sec(x)^3)
spyros
 2 years ago
integral(sec(x)^3)

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experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1\[ \int \sec^3x dx \text{ or } \int \sec (x^3) dx ??\]

spyros
 2 years ago
Best ResponseYou've already chosen the best response.0\[\int\limits(\sec(x)^3dx\]

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1\[ \sec^3x = \sec^2 \sec x = (1 + \tan^2x) \sec x\]

spyros
 2 years ago
Best ResponseYou've already chosen the best response.0actually( \[\int\limits{\sqrt(1+(x1)^2)dx}\]

spyros
 2 years ago
Best ResponseYou've already chosen the best response.0@experimentX i'll try that one

spyros
 2 years ago
Best ResponseYou've already chosen the best response.0i came to integral(sec(x)^3dx) from integral(sqrt(1+(x1)^2)dx

slaaibak
 2 years ago
Best ResponseYou've already chosen the best response.1\[\int\limits_{}^{} \sec^3(x) = \int\limits_{}^{}\sec x {d \over dx} \tan x = \sec x \tan x  \int\limits \sec (x)\tan^2x \] \[\int\limits \sec^3(x) = \sec x \tan x  \int\limits \sec x (\sec^2(x)  1) \]

slaaibak
 2 years ago
Best ResponseYou've already chosen the best response.1Now take the second part further

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1int sqrt(1 + (x1)^2) dx = (1/4*(2*x2))*sqrt(2+x^22*x)+(1/2)*arcsinh(x1)

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1if case of any confusion plugin integration to wolframalpha and click for show steps

Callisto
 2 years ago
Best ResponseYou've already chosen the best response.2Just a little try, not sure if it is correct but hope it helps \[\int sec^3xdx\]\[=\int secx(1+tan^2x)dx\]\[=\int secx+secxtan^2xdx\]\[=\int secxdx+\int secxtan^2xdx\]\[=\int \frac{secx(secx+tanx)}{secx+tanx}dx+\int tanxd(secx)\]\[=\int \frac{1}{secx+tanx}d(secx+tanx)+[secxtanx\int secxd(tanx)]\]\[=\ln secx+tanx+[secxtanx\int secxsec^2xdx] +C\]\[=\ln secx+tanx+[secxtanx\int sec^3xdx] +C\] So, we've got \[\int sec^3xdx=\ln secx+tanx+secxtanx\int sec^3xdx +C\]\[2\int sec^3xdx=\ln secx+tanx+secxtanx +C\]\[\int sec^3xdx=\frac{1}{2}(\ln secx+tanx+secxtanx) +C\]
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