∫(√a2 - x2) dx = (x/2)(√a2 - x2) + (a2/2) sin-1(x/a) + c
∫(√x2-a2) dx = (x/2)(√x2-a2)-(a2/2) log (x+√(x2-a2) + c
∫(√x2+a2) dx = (x/2)(√x2+a2)+(a2/2) log (x+√(x2+a2) + c
Question 1 :
Evaluate the following with respect to "x".
√(6 - x)(x - 4)
Solution :
√(6 - x)(x - 4) dx
(6 - x)(x - 4) = 6x - 24 - x2 + 4x
= -x2 + 10x - 24
= -[x2 - 10x + 24]
= -[x2 - 2x(5) + 52 - 52 + 24]
= -[(x-5)2 - 25 + 24]
= -[(x-5)2 - 1]
= 12 - (x-5)2
∫(√a2-x2) dx = (x/2)(√a2-x2)+(a2/2) sin-1(x/a) + c
∫√(-x2 + 10x - 24) dx = ∫√[12 + (x-5)2] dx
= ((x-5)/2)√(-x2 + 10x - 24)+(1/2)sin-1((x-5)/1) + c
Question 2 :
Evaluate the following with respect to "x".
√[9 - (2x + 5)2]
Solution :
∫√[9 - (2x + 5)2] dx
√[9 - (2x + 5)2] = √[32 - (2x + 5)2] dx
∫(√a2-x2)dx=(x/2)(√a2-x2)+(a2/2) sin-1(x/a)+c
= (2x + 5)/2√[9 - (2x + 5)2] + (9/2) sin-1[(2x + 5)/3] + c
Question 3 :
Evaluate the following with respect to "x".
√[81 + (2x + 1)2]
Solution :
∫√[81 + (2x + 1)2] dx
√[81 + (2x + 1)2] = √[92 + (2x + 1)2] dx
∫(√x2+a2) dx = (x/2)(√x2+a2)+(a2/2) log (x+√(x2+a2)+c
= ((2x+1)/3)√[92+(2x+1)2]+(9/2) log (2x+1)+√[92+ (2x+1)2]
= ((2x+1)/3)√[81 + (2x + 1)2]+(9/2) log (2x+1)+√[81 + (2x + 1)2] + c
Question 4 :
Evaluate the following with respect to "x".
√[(x + 1)2 - 4]
Solution :
∫ √[(x + 1)2 - 4] dx
∫ √[(x + 1)2 - 4] = √[(x - 1)2 - 22] dx
∫(√x2-a2)dx=(x/2)(√a2-x2)-(a2/2)log(x+ √a2-x2) + c
= ((x-1)/2)√[(x + 1)2 - 4] - (4/2) log (x - 1 - √[(x + 1)2 - 4] + c
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