In this page integration worksheet4 solution8 we are going to see solution of some practice question from the worksheet of integration.

**Question 28**

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

**Solution:**

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

= ∫ 1/[√(ax + b) - √(ax + c)] dx

= ∫ 1/[√(ax + b) - √(ax + c)] **x** [√(ax + b) + √(ax + c)] dx

= ∫ [√(ax + b) + √(ax + c)]/[√(ax + b)² - √(ax + c)²] dx

= ∫ [√(ax + b) + √(ax + c)]/[ax + b - ax - c] dx

= ∫ [√(ax + b) + √(ax + c)]/(b-c) dx

= [1/(b-c)]∫ [(ax + b)^(1/2) + (ax + c)^(1/2)] dx

= [1/(b-c)] [(ax + b)^(3/2)/(3a/2) + (ax + c)^(3/2)/(3a/2)] + C

= [1/(b-c)] [(2/3a)(ax + b)^(3/2) + (2/3a)(ax + c)^(3/2)] + C

= [2/3a(b-c)] [(ax + b)^(3/2) + (ax + c)^(3/2)] + C

**Question 29**

Integrate the following with respect to x, (x + 1) √(x + 3)

**Solution:**

= ∫ (x + 1) √(x + 3) dx

Let us consider the term which is inside the radical as "t"

t = x + 3

x = t - 3

differentiating with respect to x on both sides

dt = 1 dx

= ∫(t - 3 + 1)√t dt

= ∫(t - 2 )t^(1/2) dt

= ∫(t^[1+(1/2)] - 2 t^(1/2)) dt

= ∫[(t^(3/2)]/(3/2) - 2 t^(1/2)) dt

= [(t^(5/2)]/(5/2) - 2 [t^(3/2)]/(3/2) + C

= [(2/5)(t^(5/2)] - 2(2/3)[t^(3/2)] + C

= [(2/5)(x+3)^(5/2)] - (4/3)[(x+3)^(3/2)] + C

**Question 30**

Integrate the following with respect to x,(x - 4) √(x + 7)

**Solution:**

= ∫(x - 4) √(x + 7) dx

Let us consider the term which is inside the radical as "t"

t = x + 7

x = t - 7

differentiating with respect to x on both sides

dt = 1 dx

= ∫(t - 7 - 4)√t dt

= ∫(t - 11)t^(1/2) dt

= ∫(t^[1+(1/2)] - 11 t^(1/2)) dt

= ∫[(t^(3/2)]/(3/2) - 11 t^(1/2)) dt

= [(t^(5/2)]/(5/2) - 11 [t^(3/2)]/(3/2) + C

= [(2/5)(t^(5/2)] - 11(2/3)[t^(3/2)] + C

= [(2/5)(x+7)^(5/2)] - (22/3)[(x+7)^(3/2)] + C

integration worksheet4 solution8 integration worksheet4 solution8

- 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