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integral(sec^3 X)

thanks... i just need that for a more difficult problem... so i don't want to spend a lot on this simple integral, thanks.

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- Thread starter Gale
- Start date

- #1

- 664

- 2

integral(sec^3 X)

thanks... i just need that for a more difficult problem... so i don't want to spend a lot on this simple integral, thanks.

- #2

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- #3

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let [tex] I = \int sec^3xdx [/tex]

and let

[tex] u = secx \ \ \Rightarrow \ \ du = secxtanxdx [/tex]

[tex] dv = sec^2x \ \ \Rightarrow \ \ v = tanx [/tex]

using integration by parts we have

[tex] I = uv - \int vdu [/tex]

[tex] \Rightarrow \ \ I = secxtanx - \int secxtan^2xdx [/tex]

using the trigonometric identity tan^2 = sec^2 -1 the integral on the right side becomes

[tex] I = secxtanx - \int[secx(sec^2x - 1)]dx [/tex]

[tex] \Rightarrow I = secxtanx - \int sec^3xdx + \int secxdx [/tex]

[tex] \Rightarrow I = secxtanx - I + ln|tanx + secx| [/tex]

[tex] \Rightarrow 2I = secxtanx + ln|tanx + secx| [/tex]

[tex] \Rightarrow I = \frac{1}{2}[ \ secxtanx + ln|tanx + secx| \ ] + C[/tex]

and let

[tex] u = secx \ \ \Rightarrow \ \ du = secxtanxdx [/tex]

[tex] dv = sec^2x \ \ \Rightarrow \ \ v = tanx [/tex]

using integration by parts we have

[tex] I = uv - \int vdu [/tex]

[tex] \Rightarrow \ \ I = secxtanx - \int secxtan^2xdx [/tex]

using the trigonometric identity tan^2 = sec^2 -1 the integral on the right side becomes

[tex] I = secxtanx - \int[secx(sec^2x - 1)]dx [/tex]

[tex] \Rightarrow I = secxtanx - \int sec^3xdx + \int secxdx [/tex]

[tex] \Rightarrow I = secxtanx - I + ln|tanx + secx| [/tex]

[tex] \Rightarrow 2I = secxtanx + ln|tanx + secx| [/tex]

[tex] \Rightarrow I = \frac{1}{2}[ \ secxtanx + ln|tanx + secx| \ ] + C[/tex]

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- #4

tony873004

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- #6

xanthym

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tony873004 said:That's a great link

Is there one that does derivatives?

"(x + 3*exp[5*x])*(x*sin[x])^3"

~~

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- #7

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tony873004 said:That's a great link

Is there one that does derivatives?

Check this out:

http://www.compute.uwlax.edu/calc/

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