Integration Worksheet Solution3





In this page integration worksheet solution3 we are going to see solution of some practice questions of the integration worksheet.

Question 7

Integrate the following with respect to x      20 e^x + 3 x² - 2 cos x

Solution:

Formula :

∫ xⁿ dx = x⁽ⁿ ⁺ ¹⁾/(n + 1) + c

∫ cos x dx = sin x

∫ e^ x dx = e^x

                         = ∫ (20 e^x + 3 x² - 2 cos x) dx

                         = ∫ (20 e^x) dx + ∫ (3 x²) dx - ∫ 2 cos x dx

                         = 20 ∫ (e^x) dx + 3 ∫ (x²) dx - 2 ∫ cos x dx

                         = 20 e^x + 3 x³/3 - 2 sin x

                         = 20 e^x + x³ - 2 sin x + C

Answer is 20 e^x + x³ - 2 sin x + C 


Question 8

Integrate the following with respect to x   1/(1+Sin x)

Solution:

Formula :

(a +b)² = a² + 2 a b + b²

∫ sin x dx = - cos x + C

∫ cos x dx = sin x + C

                     = ∫ (Sin x/2 + cos x/2)² dx

                     = ∫ (Sin² x/2 + Cos² x/2 + 2 Sin x/2 Cos x/2) dx

                     = ∫ (1 + Sin 2 (x/2)) dx

                     = ∫ (1 + Sin x) dx

                     = ∫ 1 dx + ∫ Sin x dx

                     = x - Cos x + C

Answer is x - Cos x + C 


Question 9

Integrate the following with respect to x    1/(1 + Sin x)

Solution:

Formula :

(a² - b²) = (a + b) (a - b)

∫ sin x dx = - cos x + C

                     = ∫ 1/(1 + Sin x) dx

Now we are going to multiply by conjugate of (1 + Sin x)

                     = ∫ (1 - Sin x)/[(1 - Sin x) (1 + Sin x)] dx

                     = ∫ (1 - Sin x)/[(1² - Sin² x)] dx

                     = ∫ (1 - Sin x)/Cos² x dx

                     = ∫ (1/Cos² x - Sin x/Cos² x) dx

                     = ∫ Sec² x - (Sin x/Cos x) x (1/cos x) dx

                     = ∫ (Sec² x - tan x x sec x) dx

                     = ∫ Sec² x dx - ∫ tan x x sec x dx

                     = tan x - Sec x + C

Answer is tan x - Sec x + C

integration worksheet solution3 integration worksheet solution3