Integrate the following with respect to x : eαβ xα-1 e-βxα

Integrate the following with respect to x :

`alpha beta x^(alpha - 1) "e"^(- beta x^alpha)`

Sum

`int alpha beta x^(alpha - 1) "e"^(- beta x^alpha)`

Put β x^α = u

α β x^α-1 dx = du

`int alpha beta x^(alpha - 1) "e"^(- beta x^alpha) * "d"x = int "e"^(- betax^alpha) (alpha beta) x^(alpha - 1) * "d"x`

= `int "e"^(- "u") * "du"`

= `("e"^(- "u"))/(- 1) + "c"`

`int alpha beta x^(alpha - 1) "e"^(- beta x^alpha) * "d"x = - "e"^(- beta x^alpha) + "c"`

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Chapter 11: Integral Calculus - Exercise 11.6 [Page 206]

Chapter 11 Integral Calculus
Exercise 11.6 | Q 12 | Page 206

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.

Evaluate:`int(tansqrtx)/sqrtxdx`

Evaluate : `int _0^1 ("x" . ("sin"^-1 "x")^2)/sqrt (1 - "x"^2)` dx

Integrate the following functions with respect to x :

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

Integrate the following with respect to x :

x(1 – x)¹⁷

Integrate the following with respect to x :

sin⁵x cos³x

Integrate the following with respect to x:

`"e"^(2x) sinx`

Integrate the following with respect to x:

`"e"^x (tan x + log sec x)`

Integrate the following with respect to x:

`"e"^x sec x(1 + tan x)`

Find the integrals of the following:

`1/((x + 1)^2 - 25)`

Find the integrals of the following:

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

Integrate the following functions with respect to x:

`sqrt(9 - (2x + 5)^2`

Integrate the following functions with respect to x:

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