Integrate the following with respect to x: eetan-1x(1+x+x21+x2)

Question

Integrate the following with respect to x:

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

Sum

Solution

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

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

Put tan^–1 x = t

`1/(x1 + x^2) "d"x` = dt

x = tan t

∴ I = `int "e"^"t" [1 + tan "t" + tan^2"t"] "dt"`

= `int "e"^"t" [1 + tan^2"t" + tan "t"] "dt"`

= `int "e"^"t" [sec^2"t" + tan "t"] "dt"`

= `int "e"^"t" [tan "t" + sec^2"t"] "dt"`

f(x) = tan t

f'(x) = sec²t

`[int "e"^x ["f"(x) + "f"(x)] "d"x = "e"^x "f"(x) + "c"]`

∴ I = e^t tan t + c

I = `"e"^(tan^-1 (x)) * x + "c"`

I = `"e"^(tan^-1(x)) + "c"`

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Indefinite Integration with Standard Indefinite Integral Formulae - Methods of Integration

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Q 5 Q 4 Q 6

Chapter 11: Integral Calculus - Exercise 11.9 [Page 213]

APPEARS IN

Samacheer Kalvi Mathematics - Volume 1 and 2 [English] Class 11 TN Board

Chapter 11 Integral Calculus
Exercise 11.9 | Q 5 | Page 213

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