Problem 1 :
Evaluate
∫1/(1+9x2) dx
Solution :
= ∫1/(1+32x2) dx
The given exactly matches with the formula
∫1/(a2+x2) dx = 1/a tan-1(x/a) + c
= 1/(1+(3x)2) dx
= 1/3 tan-1(3x/1) + C
= 1/3 tan-1(3x) + C
Problem 2 :
Evaluate
∫1/(1-9x2) dx
Solution :
= ∫1/(1-32x2) dx
The given exactly matches with the formula
∫1/(a2-x2) dx = (1/2a) [log (a+x)/(a-x)] + c
a = 1 and x = 3x
= (1/2) [log(1+3x)/(1-3x)] + C
Problem 3 :
Evaluate
∫1/(1+x2/16) dx
Solution :
= ∫1/((1+(x/4)2) dx
The given exactly matches with the formula
∫1/(a2+x2) dx = (1/a) tan-1 (x/a) + c
a = 1 and x = x/4
= 1 tan-1 ((x/4)/1) + C
= tan-1 (x/4) + C
Problem 4 :
Evaluate
∫1/((x+2)2-4) dx
Solution :
= ∫1/((x+2)2-22) dx
The given exactly matches with the formula
∫1/(x2-a2) dx = (1/2a) [log (x-a)/(x+a)] + c
x = x+2 and a = 2
= (1/2⋅2) [log(x+2-2)/(x+2+2)] + C
= (1/4) [log(x/(x+4))] + C
Problem 5 :
Evaluate
∫1/√(25-x2) dx
Solution :
= ∫1/√(52-x2) dx
The given exactly matches with the formula
∫1/√(a2-x2) dx = sin-1(x/a) + c
a = 5 and x = x
= sin-1(x/5) + c
Problem 6 :
Evaluate
1/√(4x2-25) dx
Solution :
= ∫ 1/√((2x)2-52) dx
The given exactly matches with the formula
∫1/√(x2-a2) dx = log[x+√(x2-a2)]+C
a = 5 and x = 2x
= (1/2) log[2x+√(4x2-25)] + C
Problem 7 :
Evaluate
∫1/√(9x2+16) dx
Solution :
= ∫1/√((3x)2+42) dx
The given exactly matches with the formula
∫1/√(a2+x2) dx = log[x+√(a2+x2)] + c
x = 3x and a = 4
= log[3x+√(9x2+42) + C
= log[3x+√(9x2+16) + C
Problem 8 :
Evaluate
∫1/((3x+5)2+4) dx
Solution :
= ∫1/((3x+5)2+4) dx
The given exactly matches with the formula
∫1/(x2+a2) dx = (1/a) tan-1(x/a) + c
x = 3x+5 and a = 2
= (1/2) [tan-1(3x+5)/2] + C
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