Integration Worksheet2 Solution9

In this page integration worksheet2 solution9 we are going to see solution of some practice question from the worksheet of integration.

Question 9

(i) Integrate the following with respect to x , 1/cos² (px + a)

Solution:

∫ 1/cos² (px + a) dx

1/cos² x can be written as sec² x

∫ 1/cos² (px + a) dx =  ∫sec² (px + a) dx

∫ sec² (a x + b) dx =  (1/a) tan (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 p

So we get,     

∫sec²(p x + a) dx  = (1/p) tan (p x + a) + C

                         = tan (p x + a)/p + C


(ii) Integrate the following with respect to x , 1/sin² (L - m x)

Solution:

∫ 1/sin² (L - m x) dx

1/sin² (L - m x) can be written as cosec² x

∫ 1/sin² (L - m x) dx =  ∫cosec² (L - m x) dx

∫ cosec² (a x + b) dx =  -(1/a) cot (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 -m

So we get,     

∫cosec² (L - m x) dx = - (1/-m) cot (L - m x) + C

                            = cot (L - m x)/m + C


(iii) Integrate the following with respect to x , (a x +b)⁻⁸

Solution:

∫ (a x +b)⁻⁸ dx

∫ x^n dx =   x^(n + 1)/(n + 1) + C

Now we are going to integrate the given question by using this formula in the question instead of "n" we have -8 and instead of "a" we have "a" itself

So we get,     

∫(a x +b)⁻⁸ dx = (1/a) (a x + b)^(-8 + 1)/(-8 + 1) + C

                    = (1/a) (a x + b)^(-7)/(-7) + C

                     = -(a x + b)^-7/(7a) + C


(iv) Integrate the following with respect to x , (3 - 2 x)⁻¹

Solution:

∫ (3 - 2 x)⁻¹ dx = ∫ 1/(3 - 2 x) dx

∫ 1/(a x + b) dx =   (1/a) log (a x + b) + C

Now we are going to integrate the given question by using this formula in the question instead of "n" we have -1 and instead of "a" we have "-2".

So we get,     

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

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


(v) Integrate the following with respect to x ,  e^-x

Solution:

∫  e^-x dx = e^-x/(-1) + C

              = - e^-x + C

integration worksheet2 solution9 integration worksheet2 solution9