Integration Worksheet2 Solution10

In this page integration worksheet2 solution10 we are going to see solution of some practice question from the worksheet of integration.

Question 10

(i) Integrate the following with respect to x , tan (3 - 4 x)/cos (3 - 4 x)

Solution:

∫ tan (3 - 4 x)/cos (3 - 4 x) dx

Let us split tan (3 - 4 x)/cos (3 - 4 x) as tan (3 - 4 x) x [1/cos(3 - 4 x)]

∫ tan (3 - 4 x)/cos (3 - 4 x) dx = ∫tan (3 - 4 x) sec (3 - 4 x) dx

The given question exactly matches the formula 

∫ sec (a x + b) tan (a x + b) =   (1/a) sec (a x + b) + C

Now we are going to integrate the given question by using this formula in the question instead of "a" we have -4

So we get,     

∫tan (3 - 4 x) sec (3 - 4 x) dx = -(1/-4) sec (3 - 4 x) + C

                                          = -sec (3 - 4 x)/4 + C


(ii) Integrate the following with respect to x , 1/e^(p + q x)

∫ 1/e^(p + q x) dx = ∫ e^-(p + q x) dx

                          = e^-(p + q x)/-q + C

                          = -1/q e^(p + q x) + C


(iii) Integrate the following with respect to x ,1/tan (2 x + 3) sin (2 x + 3)

Solution:

∫  1/tan (2 x + 3) sin (2 x + 3) dx

Let us split 1/tan (2 x+3) sin (2 x+3) as 1/tan (2 x+3) x 1/sin (2 x+3)

Now we are going to write 1/tan (2 x+3) as cot (2 x + 3) and 1/sin(2x+3) as cosec (2x + 3)

∫ 1/tan (2 x + 3) sin (2 x + 3) dx = ∫cot (2 x + 3) cosec (2 x + 3) dx

The given question exactly matches the formula 

∫ cosec (a x + b) cot (a x + b) =   -(1/a) cosec (a x + b) + C

Now we are going to integrate the given question by using this formula in the question instead of "a" we have 2

So we get,     

∫cot (2 x + 3) cosec (2 x + 3) dx = -(1/2) cosec (2 x + 3) + C

                                              = -cosec (2 x + 3)/2 + C


(iv) Integrate the following with respect to x ,(L x + m) ^(1/2)

Solution:

∫ (L x + m) ^(1/2) dx

The given question exactly matches the formula 

∫ (a x + b)^(1/2) dx =  (1/a) (a x + b)^(3/2)/(3/2) + C

Now we are going to integrate the given question by using this formula in the question instead of "a" we have L

So we get,     

∫(L x + m) ^(1/2) dx = (1/L) (L x + m)^(3/2)/(3/2) + C

                             = (2/3L) (L x + m)^(3/2) + C


(iv) Integrate the following with respect to x ,√(4 - 5 x)

Solution:

∫ √(4 - 5 x) dx

The given question exactly matches the formula 

∫ (a x + b)^(1/2) dx =  (1/a) (a x + b)^(3/2)/(3/2) + C

Now we are going to integrate the given question by using this formula in the question instead of "a" we have -5

So we get,     

∫√(4 - 5 x) dx = (1/-5) (4 - 5 x)^(3/2)/(3/2) + C

                    = (-2/15) (4 - 5 x)^(3/2) + C

integration worksheet2 solution10 integration worksheet2 solution10