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