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Evaluate the following: ed∫etan-1x(1+x+x21+x2)dx - Mathematics

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प्रश्न

Evaluate the following:

`int "e"^(tan^-1x) ((1 + x + x^2)/(1 + x^2)) "d"x`

योग
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उत्तर

Let I = `int "e"^(tan^-1x) ((1 + x + x^2)/(1 + x^2)) "d"x`

Put `tan^-1x` = t

⇒ `1/(1 + x^2) * "d"x` = dt

= `int "e"^"t" (1 + tan "t" + tan^2 "t")"dt"`

= `int "e"^"t" (sec^2 "t" + tan "t")"dt"`

Here f(t) = tan t

∴ f'(t) = sec2t

= `"e"^"t" * "f"("t")`

= `"e"^"t" tan "t"`

= `"e"^(tan^-1x) * x + "c"` ....`[because int "e"^x ["f"(x) + "f'"(x)]"d"x = "e"^2"f"(x) + "C"]`

Hence, I = `"e"^(tan^-1x) * x + "C"`.

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अध्याय 7: Integrals - Exercise [पृष्ठ १६६]

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एनसीईआरटी एक्झांप्लर Mathematics [English] Class 12
अध्याय 7 Integrals
Exercise | Q 39 | पृष्ठ १६६

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