Integrate the following functions with respect to x : aacos2x-cos2acosx-cosa

Integrate the following functions with respect to x :

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

Sum

`int((cos2x - cos2 alpha)/(cosx - cos alpha)) "d"x`

= `int [((2 cos^2x - 1) - (2cos^2alpha - 1))/(cosx - cosalpha)]* "d"x`

cos 2x = cos²x – sin²x – 1

cos 2x = 2 cos²x – 1

= `int (2cos^2x - 1 - 2cos^2alpha + 1)/(cosx - cosalpha) * "d"x`

= `int (2cos^x - 2cos^2alpha)/(cosx - cosalpha) * "d"x`

= `int (2(cos^2x - cos^2alpha))/(cosx - cosalpha) * "d"x`

= `2 int [((cosx + cosalpha)(cosx - cosalpha))/(cosx - cosalpha)] * "d"x`

= `2 int(cosx + cosalpha) "d"x`

= `2 int cos x "d"x + 2 int cos alpha "d"x`

= `2 int cos x "d"x + 2 int cos alpha int "d"x`

= 2 sin x + 2 cos α(x) + c

= 2 sin x + 2x cos α + c

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

Chapter 11 Integral Calculus
Exercise 11.5 | Q 5 | Page 202

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

Integrate the following functions with respect to x :

cos 3x cos 2x

Integrate the following functions with respect to x :

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

Integrate the following with respect to x :

`x^2/(1 + x^6)`

Integrate the following with respect to x :

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