∫ dx/(a2-x2) = (1/2a) log [(a + x)/(a - x)] + c
∫ dx/(x2-a2) = (1/2a) log [(x - a)/(x + a)] + c
∫ dx/(a2+ x2) = (1/a) tan-1 (x/a) + c
∫ dx/(√a2- x2) = sin-1 (x/a) + c
∫ dx/(√x2- a2) = log (x + (√x2- a2)) + c
∫ dx/(√x2+ a2) = log (x + (√x2+ a2)) + c
∫ √(a2- x2) = (x/2) √(a2- x2) + (a2/2) sin-1(x/a) + c
∫ √(x2- a2) = (x/2) √(x2- a2) - (a2/2) log(x+√(x2- a2))+c
∫ √(x2+a2) = (x/2) √(x2+ a2) + (a2/2) log(x+√(x2+ a2))+c
Question 1 :
Integrate the following with respect to x:
1/[(x + 1)2 - 25]
Solution :
= ∫1/[(x + 1)2 - 25] dx
= ∫1/[(x + 1)2 - 52] dx
∫1/(x2 - a2) dx = (1/2a) log [(x-a)/(x+a)] + c
Here x = x + 1 and a = 5
= (1/2(5)) log [(x + 1 - 5) / (x + 1 + 5)] + c
= (1/10) log [(x - 4) / (x + 6)] + c
Question 2 :
Integrate the following with respect to x:
1/√(x2 + 4x + 2)
Solution :
= ∫1/√(x2 + 4x + 2) dx
x2 + 4x + 2 = x2 + 2x(2) + 22 - 22 + 2
= (x + 2)2 - 4 + 2
x2 + 4x + 2 = (x + 2)2 - 2
= (x + 2)2 - √22
1/√[(x + 2)2 - √22]
∫ dx/(√x2- a2) = log (x + (√x2- a2)) + c
Here x = x + 2 and a = √2
= log (x + 2 + √[(x + 2)2 - √22) + c.
= log (x + 2 + √(x2 + 4x + 2) + c
Question 3 :
Integrate the following with respect to x:
1/√((2+x)2 - 1)
Solution :
= ∫ 1/√((2+x)2 - 1) dx
∫ dx/(√x2- a2) = log (x + (√x2- a2)) + c
Here x = 2 + x and a = 1
= log (2 + x + √[(2 + x)2 - 12) + c.
= log (x + 2 + √( (x + 2)2 - 1 ) + c
Question 4 :
Integrate the following with respect to x:
1/√(x2 - 4x + 5)
Solution :
= ∫ 1/√(x2 - 4x + 5) dx
x2 - 4x + 5 = x2 - 2x(2) + 22 - 22 + 5
= (x - 2)2 - 4 + 5
x2 - 4x + 5 = (x - 2)2 + 1
= (x - 2)2 + 1
1/√(x - 2)2 + 1
∫ dx/(√x2+ a2) = log (x + (√x2+ a2)) + c
Here x = x - 2 and a = 1
= log (x - 2 + √[(x - 2)2 - 12) + c.
= log (x - 2 + √(x2 - 4x + 5)) + c
Question 5 :
Integrate the following with respect to x:
1/√(9 + 8x - x2)
Solution :
= ∫ 1/√(9 + 8x - x2)
dx
9 + 8x - x2 = - (x2 - 8x - 9)
= -(x2 - 2x(4) + 42 - 42 - 9)
= - [(x - 4)2 - 16 - 9]
9 + 8x - x2 = -[(x - 4)2 - 25]
= 52- (x - 4)2
1/√[ 52- (x - 4)2]
∫ dx/(√a2- x2) = sin-1(x/a) + c
Here x = x - 4 and a = 5
= sin-1 [(x - 4)/5] + c
= sin-1 [(x - 4)/5] + c
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