In this page integration worksheet5 solution8 we are going to see
solution of some practice question from the worksheet of integration.
Question 27
Integrate the following with respect to x,x (x - a)^m
Solution:
= ∫ x (x - a)^m dx
let t = x - a
differentiating with respect to "x"
dt = dx - 0
dt = dx
x = t + a
= ∫ x (x - a)^m dx
= ∫ (t + a) t^(m) (dt)
= ∫ [t^(m + 1) - a t^(m)] dt
= ∫ [t^(m + 1) - at^(m)] dt
= ∫ t^(m + 1) dt - ∫ at^(m) dt
= t^(m + 2)/(m + 2) - a t^(m + 1)/(m + 1) + C
now we are going to apply the value of t
= (x - a)^(m + 2)/(m + 2) - a t^(m+1)/(m+1) + C
Question 28
Integrate the following with respect to x, x² (2 - x)^15
Solution:
= ∫ x (x - a)^m dx
let t = x - a
differentiating with respect to "x"
dt = dx - 0
dt = dx
x = t + a
= ∫ x (x - a)^m dx
= ∫ (t + a) t^(m) (dt)
= ∫ [t^(m + 1) - a t^(m)] dt
= ∫ [t^(m + 1) - at^(m)] dt
= ∫ t^(m + 1) dt - ∫ at^(m) dt
= t^(m + 2)/(m + 2) - a t^(m + 1)/(m + 1) + C
now we are going to apply the value of t
= (x - a)^(m + 2)/(m + 2) - a t^(m+1)/(m+1) + C
Question 29
Integrate the following with respect to x, sin√x/√x
Solution:
= ∫ sin√x/√x dx
let t = √x
differentiating with respect to "x"
dt = (1/2√x) dx
2 dt = (1/√x) dx
= ∫ sin√x/√x dx
= ∫ sin t (2 dt)
= 2∫ sin t dt
= 2 (- cos t) + C
= - 2 cos t + C
now we are going to apply the value of t
= -2 cos √x + C
integration worksheet5 solution8 integration worksheet5 solution8