Integration Worksheet4 solution7

In this page integration worksheet4 solution7 we are going to see solution of some practice question from the worksheet of integration.

Question 25

Integrate the following with respect to x, cos p x cos q x

Solution:

We are going to multiply and divide the given question by 2

      = ∫ (1/2) 2 cos p x cos q x dx

now we are going to us the formula for 2 cos A cos B that is cos (A+B)+ cos(A-B)

      = ∫ (1/2) [ cos (p x + q x) + cos (p x - q x) ]dx

      = ∫ (1/2) [cos (p + q)x  + cos (p - q)x ]dx

      = (1/2) ∫[cos (p + q)x] dx + ∫[cos (p - q)x ] dx

      = (1/2) [sin (p+q)x/(p+q)+sin(p-q)x/(p-q)] + C


Question 26

Integrate the following with respect to x, cos² 5 x sin 10 x

Solution:

We are going apply the formula for cos² x

cos² x = (1 + cos 2x)/2

         = ∫cos² 5 x sin 10 x dx

         = ∫cos² 5 x sin 10 x dx

now we are going to us the formula for 2 cos A cos B that is cos (A+B)+ cos(A-B)

      = ∫ (1/2) [ cos (p x + q x) + cos (p x - q x) ]dx

      = ∫ (1/2) [cos (p + q)x  + cos (p - q)x ]dx

      = (1/2) ∫[cos (p + q)x] dx + ∫[cos (p - q)x ] dx

      = (1/2) [sin (p+q)x/(p+q)+sin(p-q)x/(p-q)] + C


Question 27

Integrate the following with respect to x, 1/[√(x + 1) - √(x - 2)]

Solution:

Now we are going to multiply by the conjugate of the denominator

       = ∫ 1/[√(x + 1) - √(x - 2)] dx

       = ∫ 1/[√(x + 1) - √(x - 2)] x [√(x + 1) + √(x - 2)] dx

       = ∫ [√(x + 1) + √(x - 2)]/[√(x + 1)² - √(x - 2)²] dx

       = ∫ [√(x + 1) + √(x - 2)]/[x + 1- x + 2] dx

       = ∫ [√(x + 1) + √(x - 2)]/3 dx

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

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

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

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

       = (2/9)[(x + 1)^(3/2) + (x - 2)^(3/2)] + C


integration worksheet4 solution7 integration worksheet4 solution7