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

- Back to worksheet
- Integration
- Substitution method
- Decomposition method
- Properties of integrals
- Integration-by parts
- Integration-of Sec³ x
- Standard integrals
- Integrating quadratic denominator
- Integration-using partial fractions
- Definite integrals