Integration Worksheet4 solution1

In this page integration worksheet4 solution1 we are going to see solution of some practice question from the worksheet of integration.

Question 1

Integrate the following with respect to x,(2 x - 5) (36 + 4 x)

Solution:

Now we are going to integrate the given function

     = ∫  (2 x - 5) (36 + 4 x) dx

     = ∫ [ 2 x (36) + 2 x (4 x) - 5 (36) - 5 (4 x) ] dx

     = ∫ [ 72 x + 8 x² - 180 - 20 x ] dx

     = ∫ [ 52 x + 8 x² - 180 ] dx

     = ∫ 52 x dx  + ∫ 8 x² dx - ∫ 180 dx

     = 52 ∫ x dx  + 8 ∫ x² dx - 180 ∫ dx

     = 52 [x^(1+1)/(1+1)] + 8 x^(2+1)/(2+1)- 180 x + C

     = 26 [x^2/2] + 8 x^3/3- 180 x + C

     = (8/3) x^3 + 26 x^2- 180 x + C


Question 2

Integrate the following with respect to x,(1 + x³)²

Solution:

Now we are going to compare the given function with the algebraic formula

(a + b)² = a² + 2 ab + b²

     = ∫  (1 + x³)² dx

     = ∫ [ 1² + 2 (1)(x³) + (x³)²] dx

     = ∫ [ 1 + 2 x³ + x⁶] dx

     = ∫ 1 dx + ∫ 2 x³ dx + ∫ x⁶ dx

     = x + 2 ∫ x³ dx + ∫ x⁶ dx

     = x + 2 x^(3+1)/(3+1) + x^(6+1)/(6+1) + C

     = x + 2 x⁴/4 + x⁷/7 + C

     = x + x⁴/2 + x⁷/7 + C

     = ( x⁷/7)+ (x⁴/2) + x + C


Question 3

Integrate the following with respect to x,(x³ + 4 x² - 3 x + 2)/x²

Solution:

Now we are going to split this as four terms

     = ∫ [(x³ + 4 x² - 3 x + 2)/x²] dx

     = ∫ [ x³/x²] dx + ∫ 4 x²/x² dx - ∫ (3 x/x²) dx + ∫(2/x²) dx

     = ∫ x dx + ∫ 4 dx - ∫ (3/x) dx + ∫(2/x²) dx

     = x^(1+1)/(1+1) + 4 x - 3 ∫(1/x) dx + 2∫x⁻² dx

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

     = x^2/2 + 4 x - 3 log x - 2x^-1 + C

     = x²/2 + 4 x - 3 log x - 2/x + C

integration worksheet4 solution1 integration worksheet4 solution1