Integration Worksheet5 solution6

In this page integration worksheet5 solution6 we are going to see solution of some practice question from the worksheet of integration.

Question 20

Integrate the following with respect to x, cos x/cos (x - a)

Solution:

     = ∫ 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 21

Integrate the following with respect to x,  √tan x/sin x cos x

Solution:

     = ∫  √tan x/sin x cos x dx

in the first step we are going to multiply by √tan x by the numerator and denominator.

     = ∫[√tan x/sin x cos x] [√tan x/√tan x] dx

     = ∫[tan x/√tan x sin x cos x] dx

     = ∫[(sin x/cos x)/√tan x sin x cos x] dx

     = ∫[(sec ² x)/√tan x]  dx

let t= tan x

differentiating with respect to "x"

dt = sec ² x dx

     = ∫[(sec ² x)/√tan x]  dx

     = ∫[1/√t]  dt

     = ∫ t^(-1/2)  dt

     = t^[(-1/2) + 1]/[(-1/2) + 1] + C

     = t^(1/2)/(1/2) + C

     = 2 t^(1/2) + C

     = 2 √t + C

     = 2 √tan x + C


Question 22

Integrate the following with respect to x, (log x)²/x

Solution:

     = ∫(log x)²/x dx

Let t = log x

differentiating with respect to "x"

     dt = (1/x) dx

     = ∫(log x)²/x dx

     = ∫t² dt

     = t^(2+1)/(2+1) + C

     = t³/3 + C

     = (log x)³/3 + C

     = (1/3)(log x)³ + C


Question 23

Integrate the following with respect to x, e^(3 log x) e^x⁴

Solution:

     = ∫e^(3 log x) e^x⁴ dx

     = ∫e^ log x³ e^x⁴ dx

     = ∫x³ e^x⁴ dx

Let t = x⁴

differentiating with respect to "x"

      dt = 4 x³ dx

      dt/4 = x³ dx

     = ∫e^t (dt/4)

     = (1/4) e^t + C

     = (1/4) e^x⁴ + C

integration worksheet5 solution6 integration worksheet5 solution6