Integration Worksheet4 solution2

In this page integration worksheet4 solution2 we are going to see solution of some practice question from the worksheet of integration.

Question 4

Integrate the following with respect to x,(x⁴ - x² + 2)/(x + 1)

Solution:

Now we are going to integrate the given function

     = ∫  (x⁴ - x² + 2)/(x + 1) dx

     = ∫ [(x³ - x²) + 2/(x+1)] dx

     = ∫(x³ - x²) dx + ∫ 2/(x+1) dx

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

     = (x⁴/4) - (x³/3) + 2 log (x + 1) + C


Question 5

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

Solution:

Now we are going to integrate the given function

     = ∫ [(1 + x)²/√x] dx

In the first step we are going to expand the numerator by using algebraic formula.

(a + b)² = a²+ 2 a b + b²

     = ∫ [(1 + 2 x + x²)/√x] dx

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

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

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

    =x^(1/2)/(1/2)+2x^(3/2)/(3/2)+x^(5/2)/(5/2)

    = 2 √x + 2 (2/3) x √x + (2/5) x²√x + C

    = 2 √x + (4/3) x √x + (2/5) x²√x + C


Question 6

Integrate the following with respect to x, (e^2x + e^-2x + 2)/e^x

Solution:

Now we are going to integrate the given function

     = ∫ [ (e^2x + e^-2x + 2)/e^x] dx

In the first step we are going to separate the numerator with the denominator.

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

     = ∫ e^(2 x - x) dx + ∫ (e^(-2 x - x) dx + ∫ 2 e^(-x) dx

     = ∫ e^(x) dx + ∫ (e^(-3 x) dx + ∫ 2 e^(-x) dx

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

     = e^(x) - e^(-3 x)/3 - 2 e^(-x) + C


Question 7

Integrate the following with respect to x,  sin² 3 x + 4 cos 4 x

Solution:

Now we are going to trigonometric formula for sin² x

sin²x = (1-cos 2x)/2

by comparing this formula with the given question we have 3x instead of x

sin² 3 x = [1- cos 2 (3x)]/2

            = (1- cos 6x)/2

            = ∫ [sin² 3 x + 4 cos 4 x] dx

            = ∫ [(1- cos 6x)/2 + 4 cos 4 x] dx

            = ∫ [(1- cos 6x)/2] dx + ∫ 4 cos 4 x dx

            = (1/2)∫[(1- cos 6x)] dx + 4 ∫ cos 4 x dx

            = (1/2)[x - (sin 6x/6)] + 4 (sin 4 x/4) + C

            = (1/2)[x - (sin 6x/6)] + sin 4 x + C

integration worksheet4 solution2 integration worksheet4 solution2