Integration Worksheet5 solution5

In this page integration worksheet5 solution5 we are going to see solution of some practice question from the worksheet of integration.

Question 15

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

Solution:

We are going to solve this problem by using substitution method. For that we are going to consider log x as "t"

t = log x

differentiating with respect to "x"

dt = (1/x) dx

     = ∫ sin (log x)/x dx

     = ∫ sin t dt

     = cos t + C

     = cos (log x) + C 


Question 16

Integrate the following with respect to x,  cot x/log sin x

Solution:

We are going to solve this problem by using substitution method. For that we are going to consider log x as "t"

t = log sin x

 dt =(1/sin x)cos x dx

dt = cot x

     = ∫ cot x/log sin x dx

     = ∫ (1/t) dx

     = log t + C

     = log (log sin x) + C


Question 17

Integrate the following with respect to x, sec⁴xtan x

Solution:

     = ∫ sec⁴xtan x dx

     = ∫ sec ³x (sec x tan x) dx

t = sec x

differentiating with respect to "x"

dt = sec x tan x dx

     = ∫ t³ dt

     = t^(3+1)/(3+1) + C

     = (t⁴/4) + C

     = (sec ⁴ x/4) + C


Question 18

Integrate the following with respect to x, tan³x sec x

Solution:

     = ∫ tan³x sec x dx

t = sec x

differentiating with respect to "x"

dt = sec x tan x dx

     = ∫ tan ²x tan x sec x dx

     = ∫ (sec ²x - 1) tan x sec x dx

     = ∫ (sec³ x tan x - tan x sec x) dx

     = ∫ sec² x sec x tan x dx - ∫ tan x sec x dx

     = ∫ t² dt - ∫ dt

     = t^(2+1)/(2+1) - t + C

     = (t⁴/4) + C

     = (sec ⁴ x/4) + C


Question 19

Integrate the following with respect to x, sin x/sin (x + a)

Solution:

     = ∫  sin x/sin (x + a) dx

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

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

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

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

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

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

integration worksheet5 solution5 integration worksheet5 solution5