Find the integrals of the following: 1(2+x)2-1

Find the integrals of the following:

`1/sqrt((2 + x)^2 - 1)`

Sum

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

Put 2 + x = t

dx = dt

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

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

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

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Chapter 11: Integral Calculus - Exercise 11.10 [Page 219]

Chapter 11 Integral Calculus
Exercise 11.10 | Q 3. (i) | Page 219

Evaluate : `int (1+logx)/(x(2+logx)(3+logx))dx`

Find the volume of solid generated by rotating the area bounded by x²+y² =36 and the lines x = 0, x = 3 about X -axis.

Integrate the following functions with respect to x :

cot²x + tan²x

Integrate the following functions with respect to x :

`(sin^2x)/(1 + cosx)`

Integrate the following functions with respect to x :

`(1 + cos 4x)/(cos x - tan x)`

Integrate the following functions with respect to x :

`(8^(1 + x) + 4^(1 - x))/2^x`

Integrate the following functions with respect to x :

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

Integrate the following with respect to x :

`1/(x log x log (log x))`

Integrate the following with respect to x:

`sin^-1 ((2x)/(1 + x^2))`

Integrate the following with respect to x:

`"e"^(- 4x) sin 2x`

Find the integrals of the following:

`1/sqrt(x^2 - 4x + 5)`

Integrate the following with respect to x:

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

Integrate the following with respect to x:

`(2x + 3)/sqrt(x^2 + 4x + 1)`

Integrate the following functions with respect to x:

`sqrt(x^2 + 2x + 10)`