Question

integral(sec(x)^3)

Answer 1

\[ \int \sec^3x dx \text{ or } \int \sec (x^3) dx ??\]

Answer 2

\[\int\limits(\sec(x)^3dx\]

Answer 3

former or latter??

Answer 4

former

Answer 5

\[ \sec^3x = \sec^2 \sec x = (1 + \tan^2x) \sec x\]

Answer 6

actually( \[\int\limits{\sqrt(1+(x-1)^2)dx}\]

Answer 7

@experimentX i'll try that one

Answer 8

i came to integral(sec(x)^3dx) from integral(sqrt(1+(x-1)^2)dx

Answer 9

\[\int\limits_{}^{} \sec^3(x) = \int\limits_{}^{}\sec x {d \over dx} \tan x = \sec x \tan x - \int\limits \sec (x)\tan^2x \]



\[\int\limits \sec^3(x) = \sec x \tan x - \int\limits \sec x (\sec^2(x) - 1) \]

Answer 10

ok, i get it

Answer 11

Now take the second part further

Answer 12

int sqrt(1 + (x-1)^2) dx = (1/4*(2*x-2))*sqrt(2+x^2-2*x)+(1/2)*arcsinh(x-1)

Answer 13

if case of any confusion plugin integration to wolframalpha and click for show steps

Answer 14

pellet

Answer 15

sh.it

Answer 16

Just 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
\[\int sec^3xdx=\ln |secx+tanx|+secxtanx-\int sec^3xdx +C\]\[2\int sec^3xdx=\ln |secx+tanx|+secxtanx +C\]\[\int sec^3xdx=\frac{1}{2}(\ln |secx+tanx|+secxtanx) +C\]