∫sec2x dx - Mathematics and Statistics

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Sum

`int sec^2x  "d"x`

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Solution

Let I = `int sec^2x  "d"x`

= `int secx * sec^2x  "d"x`

= `sec x int sec^2x  "d"x - int["d"/("d"x)(sec x) intsec^2 x  "d"x]  "d"x` 

= `secx * tanx - int secx tanx * tanx  "d"x`

= `secx * tanx - int secx tan^2x  "d"x`

= `secx * tanx - int secx(sec^2x - 1)  "d"x`

=`secx * tanx - int sec^3x  "d"x + int secx  "d"x`

∴ I = `secx * tanx - "I" + log|secx + tanx| + "c"_1`

∴ 2I  = `secxtanx + log|secx + tanx| + "c"_1`

∴ I = `1/2[secx tanx + log|secx + tanx|] + "c"` where c = `("c"_1)/2`

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Chapter 2.3: Indefinite Integration - Short Answers II

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