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Question
`int sec^3x "d"x`
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Solution
Let I = `int sec^3x "d"x`
= `int secx * sec^3x "d"x`
= `sec x int sec^3x "d"x - int["d"/("d"x)(sec x) intsec^3 x "d"x] "d"x`
= `secx * tanx - int secx tanx * tanx "d"x`
= `secx * tanx - int secx tan^3x "d"x`
= `secx * tanx - int secx(sec^3x - 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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