Integration Worksheet Solution2





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

Question 4

Integrate the following with respect to x  (x + 1)/ x²

Solution:

Formula :

∫ xⁿ dx = x⁽ⁿ ⁺ ¹⁾/(n + 1) + c


∫(1/x) dx= log x + C

                        =  ∫ (x + 1)/ x² dx

Now we are going to write separate integration symbol and dx

                        = ∫(x/x²) dx + ∫(1/x²) dx

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

                        =  log x + x⁻¹

                        =  log x + 1/x + C

Answer is log x + 1/x + C


Question 5

Integrate the following with respect to x     (x + (1/x))²

Solution:

Formula :

∫ xⁿ dx = x⁽ⁿ ⁺ ¹⁾/(n + 1) + c

(a +b)² = a² + 2 a b + b²

(x + (1/x))² = x² + (1/x²) + (2 x x x 1/x )
Now we are going to write separate integration symbol and dx

∫ (x + (1/x))² dx  = ∫ x² dx  + ∫ (1/x²) dx + ∫ (2 x x x 1/x ) dx

                       = ∫ x² dx  + ∫ x⁻² dx + ∫ 2 dx

                       = ∫ x² dx  + ∫ x⁻² dx + 2 ∫ dx

                       = (x³/3) + x⁻ ¹/(-1) + 2 x + C

                       = (x³/3) - 1/x + 2 x + C

Answer is (x³/3) - 1/x + 2 x + C


Question 6

 Integrate the following with respect to x   (1 - x)³

∫ xⁿ dx = x⁽ⁿ ⁺ ¹⁾/(n + 1) + c

(a - b)³ = a³ - 3 a² b + 3 a b² - b³

(1 - x)³ = 1³ - 3 (1)² (x) + 3 (1) x² - x³

           = 1 - 3 x + 3 x² - x³

∫ (1 - x)³ dx = ∫ (1 - 3 x + 3 x² - x³) dx

                  = ∫ 1 dx - ∫ 3 x dx + ∫ 3 x² dx - ∫ x³ dx

                  = ∫ dx - 3 ∫ x dx + 3 ∫ x² dx - ∫ x³ dx

                  = x - 3x²/2 + 3 x³/3 - x⁴/4 + C

                  = x - 3x²/2 + x³ - x⁴/4 + C

Answer is x - 3x²/2 + x³ - x⁴/4 + C

integration worksheet solution2 integration worksheet solution2