Find the integrals of the following: 19+8x-x2 - Mathematics

Find the integrals of the following:

`1/sqrt(9 + 8x - x^2)`

बेरीज

`int 1/sqrt(9 + 8x - x^2) "d"x = int ("d"x)/sqrt(9 - (x^2 - 8x))`

= `int ("d"x)/sqrt(9 - [(x - 4)^2 - 4^2]`

= `int ("d"x)/sqrt(9 - [(x - 4)^2 - 16]`

= `int ("d"x)/sqrt(9 - (x - 4)^2 + 16)`

= `int ("d"x)/sqrt(25 - (x - 4)^2`

= `int ("d"x)/sqrt(5^2 - (x - 4)^2`

Put x – 4 = t

dx = dt

`int ("d")/sqrt(9 + 8x - x^2) = int ("d"x)/sqrt(5^2 - "t"^2)`

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

`int ("d"x)/sqrt(9 + 8x - x^2) = sin^-1 ("t"/5) + "c"`

= `sin^-1 ((x - 4)/5) + "c"`

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पाठ 11: Integral Calculus - Exercise 11.10 [पृष्ठ २१९]

पाठ 11 Integral Calculus
Exercise 11.10 | Q 3. (iii) | पृष्ठ २१९

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.

Find the volume of the solid generated by the complete revolution of the ellipse `"x"^2/36 + "y"^2/25 = 1` about Y-axis.

Integrate the following functions with respect to x :

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

Integrate the following functions with respect to x :

`"e"^(x log "a") "e"^x`

Integrate the following functions with respect to x :

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

Integrate the following with respect to x :

`(sin 2x)/("a"^2 + "b"^2 sin^2x)`

Integrate the following with respect to x :

`tan x sqrt(sec x)`