Integrate the following functions with respect to x: x2-2x-3

Integrate the following functions with respect to x:

`sqrt(x^2 - 2x - 3)`

Sum

`int sqrt(x^2 - 2x - 3) "d"x = int sqrt((x - 1)^2 - 1^2 - 3) "d"x`

= `int sqrt((x - 1)^2 - 4) "d"x`

= `int sqrt((x - 1)^2 - 2^2) "d"x`

Put x – 1 = t

dx = dt

= `int sqrt("t"^2 - 2^2) "dt"`

= `"t"/2 sqrt("t"^2 - 2^2) - 2^2/2 log |"t" + sqrt("t"^2 - 2^2)| + "c"`

= `(x - 1)/2 sqrt((x - 1)^2 - 4) - 2 log|x - 1 + sqrt((x - 1)^2 - 4)| + "c"`

= `(x - 1)/2 sqrt(x^2 - 2x + 1 - 4) - 2 log|x - 1 sqrt(x^2 - 2x + 1 - 4) | + "c"`

`int sqrt(x^2 - 2x - 3) "d"x = (x - 1)/2 sqrt(x^2 - 2x - 3) - 2 log |x - 1 + sqrt(x^2 - 2x - 3)| + "c"`

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Chapter 11: Integral Calculus - Exercise 11.12 [Page 225]

Chapter 11 Integral Calculus
Exercise 11.12 | Q 1. (ii) | Page 225

Evaluate : `∫_0^(pi/2) (sinx.cosx)/(1 + sin^4x)`.dx

Integrate the following functions with respect to x :

`[sqrt(x) + 1/sqrt(x)]^2`

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cot²x + tan²x

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`(3x + 4) sqrt(3x + 7)`

Integrate the following functions with respect to x :

`x^3/((x - 1)(x - 2))`

Integrate the following with respect to x :

`("e"^x - "e"^-x)/("e"^x + "e"^-x)`

Integrate the following with respect to x:

x sec x tan x

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x⁵e^x2

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`"e"^("a"x) cos"b"x`

Integrate the following with respect to x:

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

Integrate the following with respect to x:

`(2x - 3)/(x^2 + 4x - 12)`

Integrate the following functions with respect to x:

`sqrt(81 + (2x + 1)^2`