PROBLEMS ON STANDARD INTEGRALS

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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