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Question
Evaluate : `int ("e"^"x" (1 + "x"))/("cos"^2("x""e"^"x"))"dx"`
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Solution
Let I = `int ("e"^"x" (1 + "x"))/("cos"^2("x""e"^"x"))"dx"`
Put `"x"."e"^"x" = "t"`
Differentiate w.r.t. x,
`"x" ."e"^"x" +"e"^"x" . 1 = "dt"/"dx" => "e"^"x" ("x" +1)"dx" ="dt"`
`therefore int 1/("cos"^2 "t") "dt"`
= ∫ sec2 t dt
= tan t + c
` therefore int ("e"^"x" (1 + "x"))/("cos"^2("x""e"^"x"))"dx" = "tan"("x" ."e"^"x") + "c"`
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