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Evaluation of Definite integration by method of the substitution

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Evaluation of Definite integration by method of the substitution - Lesson Summary

Steps to evaluate ab f(x) dx by the method of substitution:

01 (2x + 3)√(3 - 2x) dx


Step - 1) Consider the integral without taking the given limits.

Consider ∫ (2x + 3)√(3 - 2x) dx


Step -2) Substitute t = f(x) or x = g(t) to reduce the given integral to a known form.

Let (3 - 2x) = t2

⇒ 2x = 3 - t2

⇒ 2x + 3 = 6 - t2

2 dx = -2t dt

⇒dx = -t dt

∫ (2x + 3)√(3 - 2x) dx = ∫ (6 - t2)√t2 x (-tdt)

∫ - (6 - t2)t2 dt

= ∫ (t4 - 6t2) dt


Step -3) Integrate the new integrand with respect to the new variable without placing the constant of integration.

= (t5/5) - 6(t3/3)

= t5/5 - 6t3/3

= t5/5 - 2t3

∫ (2x + 3)√(3 - 2x) dx = [ t5/5 - 2t3]


Step - 4) Write the answer in terms of the original variable by re-substituting the new variable.

∫ (2x + 3)√(3 - 2x) dx = [ (√(3 - 2x))5/5 - 2(√(3 - 2x))3]


Step - 5) Find the values of the answers obtained in Step - 4 at the given upper and lower limits.

01 (2x + 3)√(3 - 2x) dx = [ (√(3 - 2x))5/5 - 2(√(3 - 2x))3]01


Step - 6) Subtract the value at the lower limit from the value of the upper limit to obtain the required definite integral.

= [ (√(3 - 2(1)))5/5 - 2(√(3 - 2(1)))3] - [ (√(3 - 2(0)))5/5 - 2(√(3 - 2(0)))3]

= 1/5 - 2 - 9√3/5 + 6√3

= (1 - 10 - 9√3 + 30√3)/5

= (-9 + 21√3)/5

01 (2x + 3)√(3 - 2x) dx = (-9 + 21√3)/5

Evaluate ∫ab √(x-a/b-x) dx


Step - 1) Consider the integral without taking the given limits.


Step -2) Substitute t = f(x) or x = g(t) to reduce the given integral to a known form.

Put x = a cos2 θ + b sin2 θ

dx = a(-2 cos θ sin θ) dθ + b(2 sin θ cos θ)

     = -a sin 2θ dθ + b sin 2θ dθ

     = (b - a) sin 2θ dθ

x - a = (a cos2 θ + b sin2 θ - a)

        = b sin2 θ - a(1 - cos2 θ)

        = b sin2 θ - a sin2 θ

        = (b - a) sin2 θ

b - x = (b - acos2 θ - bsin2 θ)

        = b(1 - sin2 θ) - acos2 θ

         = b cos2 θ - a cos2 θ

         = (b - a) cos2 θ

When x = a, a cos2 θ + b sin2 θ = a

⇒ b sin2 θ = a (1 - cos2 θ)

⇒ b sin2 θ = a sin2 θ

⇒ (b - a) sin2 θ = 0

As (b - a) ≠ 0, sin θ = 0

⇒ θ = 0o

∴ When x = a; θ = 0o

∴ When x = b, a cos2 θ + bsin2 θ = b

⇒ a cos2 θ = b(1 - sin2 θ)

⇒ a cos2 θ = b cos2 θ

⇒ cos2 θ (a - b) = 0

As a - b ≠ 0 , cos θ = 0

⇒ θ = π/2

∴ When x = b,; θ = π/2


Step -3) Integrate the new integrand with respect to the new variable without placing the constant of integration.

ab √(x-a/b-x) dx


Step-4) Keep the integral in the new variable itself and change the limits of the integral accordingly.

= ∫0π/2 √((b-a)sin2θ/(b-a)cos2θ) x (b - a) sin 2θ dθ

= ∫0π/2 (b-a) sin θ/cos θ x 2 sin θ cos θ dθ

= 2(b - a) ∫0π/2 sin2θ dθ

= 2(b - a) ∫0π/2 (1 - cos2θ)/2 dθ


= (b - a) [ ∫0π/2 dθ - ∫0π/2 cos2θ dθ


Step - 5) Find the values of the answers obtained in the previous step at the given upper and lower limits.

= (b - a) [ [θ]0π/2 - [sin2θ/2]0π/2 ]


Step - 6) Subtract the value at the lower limit from the value of the upper limit to obtain the required definite integral.

= (b-a) [ (π/2 - 0) - (sin(2xπ/2)/2 - sin(2x0)/2)]

= 2(b - a)[π/4 - 1/2.(0)]

= 2(b - a) x π/4

= (b - a) π/2

Hence,
ab √(x-a/b-x) dx = (b - a) π/2

Hence, using these two methods, we can evaluate a given definite integral by the method of substitution.

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