Integration Worksheet6 solution1

In this page integration worksheet6 solution1 we are going to see solution of some practice question from the worksheet of integration.

Question 1

Integrate the following with respect to x,    x e^-x

Solution:

Here we are going to use the method partial differentiation to integrate the given question.

∫ x e^-x dx

∫ u dv = u v - ∫ v du

u = x           dv = e^-x

du = dx         v = - e^-x

     = ∫ x e^-x dx

     =  x ( - e^-x) - ∫ (- e^-x) dx

    = - x e^-x + ∫ e^-x dx    

    = - x e^-x - e^-x + C

    = -e^-x (x + 1) + C


Question 2

Integrate the following with respect to x,    x cos x

Solution:

Here we are going to use the method partial differentiation to integrate the given question.

∫ x cos x dx

∫ u dv = u v - ∫ v du

u = x           dv = cos x

du = dx         v = sin x

     = ∫ x cos x dx

     =  x (sin x)  - ∫ sin x  dx

     =  x (sin x)  - (- cos x) + C

     =  x  sin x + cos x + C


Question 3

Integrate the following with respect to x,     x cosec ²x

Solution:

Here we are going to use the method partial differentiation to integrate the given question.

∫ x cosec ²x dx

∫ u dv = u v - ∫ v du

u = x           dv = cosec ²x

du = dx         v = -cot x

     = ∫ x cosec ²x dx

     =  x (-cot x) - ∫ -cot x dx

     =  - x cot x + ∫ (cos x/sin x) dx

     =  - x cot x + log (sin x) + C


Question 4

Integrate the following with respect to x,     x sec x tan x dx

Solution:

Here we are going to use the method partial differentiation to integrate the given question.

∫x sec x tan x  dx

∫ u dv = u v - ∫ v du

u = x           dv = sec x tan x

du = dx         v = sec x

     = ∫ x sec x tan x  dx

     =  x (sec x) - ∫sec x dx

     = x sec x - log (sec x + tan x) + C

integration worksheet6 solution1 integration worksheet6 solution1