Notes On Evaluation of Definite integration by method of the substitution - CBSE Class 12 Maths
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.

#### 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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