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Question
Evaluate `int x^2"e"^(4x) "d"x`
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Solution
Let I = `int x^2*"e"^(4x) "d"x`
= `x^2 int "e"^(4x) "d"x - int ["d"/("d"x)(x^2) int"e"^(4x)"d"x]"d"x`
= `x^2* ("e"^(4x))/4 - int 2x* ("e"^(4x))/4 "d"x`
= `(x^2*"e"^(4x))/4 - 1/2 intx*"e"^(4x)"d"x`
= `(x^2"e"^(4x))/4 - 1/2[x"f""e"^(4x)"d"x - int("d"/("d"x)(x) int"e"^(4x)"d"x)"d"x]`
= `(x^2"e"^(4x))/4 - 1/2[x* ("e"^(4x))/4 - int 1* ("e"^(4x))/4 "d"x]`
= `(x^2"e"^(4x))/4 - 1/2[(x*"e"^(4x))/4 - 1/4 int"e"^(4x)"d"x]`
= `(x^2"e"^(4x))/4 - 1/2[(x"e"^(4x))/4 - 1/4*("e"^(4x))/4] + "c"`
= `(x^2"e"^(4x))/4 - (x"e"^(4x))/8 + ("e"^(4x))/32 + "c"`
∴ I = `("e"^(4x))/4[x^2 - x/2 + 1/8] + "c"`
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