Integration Worksheet2 Solution2





In this page integration worksheet2 solution2 we are going to see solution of some practice question from the worksheet of integration.

Question 2

(i) Integrate the following with respect to x , 1/x⁶

Solution:

The given question exactly matches the formula 

∫ xⁿ 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 -6

So we get,     

∫ x⁻⁶ dx  = x^(-6 + 1)/(-6 + 1)

            = -(x^-5/5 + C

            = -1/5x⁵ + C  


(ii) Integrate the following with respect to x,1/ (x + 5) ⁴

Solution:

∫ 1/ (x + 5) ⁴ dx

Let us use the substitution method to integrate this. For that let us consider (x + 5) as "t"

t = x + 5

dt = 1 dx

∫ 1/ (x + 5) ⁴ dx = ∫ (1/t⁴) dt

                         = ∫ t⁻⁴ dt

                         = t^(-4 + 1)/(-4 + 1) + C

                         = t^-3/(-3) + C

                          = -1/3 t^3 + C

                          = -1/3 (x + 5)^3 + C                           


(iii) Integrate the following with respect to x,1/(2 x + 3) ⁵

Solution:

∫ 1/(2 x + 3) ⁵ dx

Let us use the substitution method to integrate this. For that let us consider (2 x + 3) as "t"

t = 2 x + 3

dt = 2 dx

dx = dt/2

∫ 1/ (2 x + 3) ⁵ dx = ∫ (1/t⁵) (dt/2)

                         = ∫ t⁻⁵ (dt/2)

                         = (1/2)t^(-5 + 1)/(-5 + 1) + C

                         = (1/2)t^-4/(-4) + C

                          = -1/8 t^4 + C

                          = -1/8 (2 x + 3)^4 + C    


(iv) Integrate the following with respect to x, 1/(4 - 5 x) ⁷

Solution:

∫  1/(4 - 5 x) ⁷ dx

Let us use the substitution method to integrate this. For that let us consider (4 - 5 x) as "t"

t = 4 - 5 x

dt = - 5 dx

dx = -dt/5

∫ 1/ (4 - 5 x)⁷ dx = ∫ (1/t⁷) (-dt/5)

                         = ∫ t⁻⁷ (-dt/5)

                         = (1/5)t^(-7 + 1)/(-7 + 1) + C

                         = (1/5)t^-6/(-6) + C

                          = -1/30 t^6 + C

                          = -1/30 (4 - 5 x)^6 + C    


(v) Integrate the following with respect to x,  1/(a x + b)⁸

Solution:

∫   1/(a x + b)⁸ dx

Let us use the substitution method to integrate this. For that let us consider (a x + b) as "t"

t = a x + b

dt = a dx

dx = dt/a

∫ 1/ (a x + b)⁸ dx = ∫ (1/t⁸ ) (dt/a)

                         = ∫ t⁻⁸ (dt/a)

                         = (1/a)t^(-8 + 1)/(-8 + 1) + C

                         = (1/a)t^-7/(-7) + C

                          = 1/7a t^7 + C

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

integration worksheet2 solution2 integration worksheet2 solution2