Integration Worksheet3 solution2

In this page integration worksheet3 solution2 we are going to see solution of some practice question from the worksheet of integration.

Question 6

Integrate the following with respect to x,4/(3+4x)+(10x+3)⁹-3cosec(2x+3) cot (2x+3)

Solution:

Now we are going to integrate the given function

     = ∫ [4/(3+4x)+(10x+3)⁹-3cosec(2x+3) cot (2x+3)] dx

     = ∫ 4/(3+4x) dx + ∫ (10x+3)⁹ dx - ∫ 3cosec(2x+3) cot (2x+3) dx

     = 4∫1/(3+4x) dx + ∫ (10x+3)⁹ dx - 3 ∫cosec(2x+3) cot (2x+3) dx

     = 4log(3+4x)/4+(10x+3)^(9+1)/10(9+1)- 3[-cosec(2x+3)]/2 + C

     = log(3+4x)+(10x+3)^10/100+(3/2)[cosec(2x+3)] + C


Question 7

Integrate the following with respect to x,6 sin 5x - 1/(p x + q)^m

Solution:

Now we are going to integrate the given function

     = ∫ [6 sin 5x - 1/(p x + q)^m] dx

     = ∫ 6 sin 5x dx - ∫ 1/(p x + q)^m dx

     = 6 (-cos 5x)/5 - ∫ (p x + q)^-m dx

     = -(6/5) cos 5x - [(p x + q)^(-m+1)/p(-m+1) + C

     = -(6cos 5x/5)  - (1/p)[(p x + q)^(1-m)/(1-m) + C


Question 8

Integrate the following with respect to x,a sec²(b x + c) + q/e^(L - m x)

Solution:

Now we are going to integrate the given function

     = ∫ [a sec²(b x + c) + q/e^(L - m x)] dx

     = ∫ a sec²(b x + c) dx + ∫ q/e^(L - m x) dx

     = a tan(b x + c)/b  + q ∫ e^-(L - m x)dx

     = (a/b) tan (b x + c) + q [e^-(L - m x)/(-m)] + C

     = (a/b) tan (b x + c) - (q/m) [1/e^(L - m x)] + C


Question 9

Integrate the following with respect to x,1/[3+(2x/3)]-(2/3)cos (x-2/3)+3(x//3 + 4)^6

Solution:

Now we are going to integrate the given function

     = ∫ [1/[3+(2x/3)]-(2/3)cos (x-2/3)+3(x//3 + 4)^6] dx

     = ∫ [1/[3+(2x/3)] dx-∫(2/3)cos (x-2/3) dx+∫3(x/3 + 4)^6] dx

     = ∫ [1/[3+(2x/3)] dx-(2/3)∫cos (x-2/3) dx+ 3∫(x/3 + 4)^6] dx

     = log [3+(2x/3)]/(2/3)-(2/3)sin (x-2/3)+ 3(x/3 + 4)^(6+1)/(1/3)(6+1)]

     = (3/2)log [3+(2x/3)]-(2/3)sin (x-2/3)+ 3(x/3 + 4)^7/(1/3)7] + C

     = (3/2)log [3+(2x/3)]-(2/3)sin (x-2/3)+(9/7)(x/3 + 4)^7 + C


Question 10

Integrate the following with respect to x, 7 sin x/7 - 8 sec²[4 - (x/4)] + 10 [(2x/5) - 4]^3/2

Solution:

Now we are going to integrate the given function

     = ∫ [ 7 sin x/7 - 8 sec²[4 - (x/4)] + 10 [(2x/5) - 4]^3/2] dx

     = ∫ [7 sin x/7] dx-∫8 sec²[4 - (x/4)] dx+∫10 [(2x/5) - 4]^3/2] dx

     = 7∫sin (x/7) dx-8 ∫sec²[4 - (x/4)] dx+ 10∫ [(2x/5) - 4]^3/2] dx

  =7[-cos (x/7)/(1/7)]-8tan[4-(x/4)]/(-1/4)+10[(2x/5)-4]^(3/2+1)/(3/2+1)]

  = -49[cos (x/7)]+32tan[4-(x/4)]+10[(2x/5)-4]^(5/2)/(5/2)(2/5)]

  = -49[cos (x/7)]+32tan[4-(x/4)]+10[(2x/5)-4]^(5/2)]

  = -49[cos (x/7)]+32tan[4-(x/4)]+10[(2x/5)-4]^(5/2)] + C

integration worksheet3 solution2 integration worksheet3 solution2