# Easy calc problem

Hey, i know this is an easy problem, but i don't have any of my older calc books with the identities, so, just hoping for a quick answer

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.

let $$I = \int sec^3xdx$$

and let

$$u = secx \ \ \Rightarrow \ \ du = secxtanxdx$$

$$dv = sec^2x \ \ \Rightarrow \ \ v = tanx$$

using integration by parts we have

$$I = uv - \int vdu$$

$$\Rightarrow \ \ I = secxtanx - \int secxtan^2xdx$$

using the trigonometric identity tan^2 = sec^2 -1 the integral on the right side becomes
$$I = secxtanx - \int[secx(sec^2x - 1)]dx$$

$$\Rightarrow I = secxtanx - \int sec^3xdx + \int secxdx$$

$$\Rightarrow I = secxtanx - I + ln|tanx + secx|$$

$$\Rightarrow 2I = secxtanx + ln|tanx + secx|$$

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

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tony873004
Gold Member
gnome said:

Is there one that does derivatives?

Derivatives are easy for all products, quotients, powers, compositions, and sums of elementary functions... antiderivatives are much more difficult.

xanthym
tony873004 said:

Is there one that does derivatives?
YES ... Integrals, Derivatives, Graphing, and Matrices ... GO TO:
-----> http://www.calc101.com/
Select from list upper right corner home page. Remember to enter functions with SQUARE BRACKETS "[]", like "sin[x]", and group with PARENTHESES "()", like:
"(x + 3*exp[5*x])*(x*sin[x])^3"

~~

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