Integrate the following functions with respect to x : 81+x+41-x2x - Mathematics

Integrate the following functions with respect to x :

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

Sum

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

= `int ((8^(1 + x))/2^x + (4^(1- x))/2^x) "d"x`

= `int ((8^1 * 8^x)/2^x + (4^1 * 4^-x)/2^2x) "d"x`

= `int (8* 8^x)/2^x "d"x + int (4* 4^-x)/2^x * "d"x`

= `8 int (2^3)^x/2^x "d"x + 4int (2^2)^x/2^x * "d"x`

= `8int 2^(3x)/2^x "d"x + 4int 2^(-2x)/2^x * "d"x`

= `8int 2^(3x) 2^(- x) * "d"x + 4int 2^(- 2x) 2^(- x) * "d"x`

= `8int 2^(3x - x) * "d"x + 4int 2^(- 2x - x) * "d"x`

= `8int 2^(2x) * "d"x + 4int 2^(- 3x) * "d"x`

= `8 xx 1/2 xx 2^(2x)/log 2 + 4 xx 1/(-3) xx 2^(- 3x)/log2 + "c"`

= `4 xx 2^(2x)/log2 - 4/3 xx 2^(- 3x)/log2 + "c"`

= `(2^(2x) xx 2^(2x))/log2 - (2^2 xx 2^(- 3x))/(3 log 2) + "c"`

= `2^(2x + 2)/log2 - 2^(2 - 3x)/(3log2) + "c"`

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Chapter 11: Integral Calculus - Exercise 11.5 [Page 202]

Chapter 11 Integral Calculus
Exercise 11.5 | Q 15 | Page 202

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.

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`(cos2x - cos 2 alpha)/(cosx - cos alpha)`

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`1/(sqrt(x + 3) - sqrt(x - 4))`

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

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`(10x^9 + 10^x log_"e" 10)/(10^x + x^10)`

Integrate the following with respect to x :

`sqrt(x)/(1 + sqrt(x))`