∫ 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/(4 - x2)
Solution :
= ∫1/(4 - x2) dx
= ∫1/(22 - x2) dx
∫1/(a2 - x2) dx = (1/2a) log [(a + x)/(a - x)] + c
= (1/2(2)) log [(2 + x) / (2 - x)] + c
= (1/4) log [(2 + x) / (2 - x)] + c
Question 2 :
Integrate the following with respect to x:
1/(25 - 4x2)
Solution :
= ∫1/(25 - 4x2) dx
= ∫1/(52 - (2x)2) dx
here a = 5 and x = 2x
= (1/2⋅2(5))log [(5 + 2x) / (5 - 2x)] + c
= (1/20)log [(5 + 2x) / (5 - 2x)] + c
Question 3 :
Integrate the following with respect to x:
1/(9x2 - 4)
Solution :
= ∫ 1/((3x)2 - 22) dx
∫ dx/(x2-a2) = (1/2a) log [(x - a)/(x + a)] + c
Here x = 3x and a = 2
= (1/2⋅3(2))log [(3x - 2) / (3x - 2)] + c
= (1/12)log [(3x - 2) / (3x - 2)] + c
Question 4 :
Integrate the following with respect to x:
1/(6x - 7 - x2)
Solution :
= ∫ 1/(6x - 7 - x2)dx
= -∫ 1/(x2 - 6x + 7)dx
x2 - 6x + 7 = x2 - 2x(3) + 32 - 32+ 7
= (x - 3)2 - 9 + 7
= (x - 3)2 - 2
x2 - 6x + 7 = (x - 3)2 - (√2)2
-∫ 1/(x2 - 6x + 7)dx = -∫ 1/[(x - 3)2 - (√2)2]dx
= ∫1/[√22 - (x - 3)2]dx
∫1/(a2 - x2) dx = (1/2a) log [(a + x)/(a - x)] + c
= (1/2√2) log [(√2 + (x - 3)) / (√2 - (x - 3))]
= (1/2√2) log [(√2 + x - 3) / (√2 - x + 3)] + c
= (1/2√2) log [(√2 - 3 + x) / (√2 + 3 - x)] + c
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