Integration Worksheet5 solution7

In this page integration worksheet5 solution7 we are going to see solution of some practice question from the worksheet of integration.

Question 24

Integrate the following with respect to x, sin 2x/(a cos²x + b sin²x)

Solution:

t = (a cos²x + b sin²x)

dt = 2 a cos x (-sin x) + 2 b sin x cos x

dt = -2 a sin x cos x + 2 b sin x cos x

dt = - a sin 2x + b sin 2x

dt = sin 2x(b - a)

dt = 2 [b sin x cos x- a sin x cos x]

     = ∫ cos x/cos (x - a) dx

     = ∫ cos (x + a - a)/cos (x - a) dx

now we are going to compare the numerator with the formula cos (A- B) = cos A cos B - sin A sin B

   = ∫ [cos (x + a) cos a - sin (x + a) sin a]/cos (x + a) dx

   = ∫[cos (x + a) cos a]/cos (x + a) dx - ∫[sin (x + a) sin a]/cos (x + a) dx

   = ∫ cos a dx - ∫tan (x + a) sin a dx

   = x cos a - sin a log cos (x - a) + C


Question 25

Integrate the following with respect to x,x (L - x)^16

Solution:

                = ∫ x (L - x)^16 dx

let t = L - x

differentiating with respect to "x"

dt = 0 - dx

dt = - dx

dx = - dt

x = L - t

                = ∫ x (L - x)^16 dx

                = ∫ (L - t) t^(16) (-dt)

                = ∫ [L t^(16) - t^(17)] (-dt)

                = ∫ [t^(17) - L t^(16)] dt

                = ∫ t^(17) dt - L∫ t^(16)dt

                = t^(17 + 1)/(17 + 1) - L t^(16 + 1)/(16 + 1) + C

                = t^(18)/(18) - L t^(17)/(17) + C

now we are going to apply the value of t

                 = (L - x)^(18)/(18) - L (L - x)^(17)/(17) + C


Question 26

Integrate the following with respect to x, (1 - tan x)/(1 + tan x)

Solution:

                = ∫ (1 - tan x)/(1 + tan x) dx

                = ∫ (tan π/4 - tan x)/(1 + tan π/4 tan x) dx

                = ∫ tan (π/4 - x) dx

                = [log sec (π/4 - x)]/(-1)

                = (-1) [log sec (π/4 - x)]

                = log [sec (π/4 - x)]⁻ ¹

                = log (1/(sec (π/4 - x))

                = log cos (π/4 - x) + C

integration worksheet5 solution7 integration worksheet5 solution7