Indefinite Integrals


   
 
Some Special Integrals
01. Prove that
 
 
Proof:
 
 
 
 
1 = A(x + a) + B (x - a)
 
 
 
 
 
 
 
02. Prove that
 
 
Proof:
 
 
 
 
A(a - x) + B(a + x)
 
When x = -a,
 
 
When x = a,
 
 
 
 
 
 
03. Prove that
 
 
Proof:
 
 
Put x = atanq
 
 
dx = asec2q dq
 
 
 
 
 
04. Prove that
 
 
Proof:
 
 
Put x = a sec q
 
dx = a secq tanq dq
 
 
 
 
 
 
 
 
05. Prove that
 
 
Proof:
 
 
Put x = a sinq
 
 
dx = a cosqdq
 
 
 
q + C
 
 
06. Prove that
 
 
 
Proof:
 
 
Put x = a tan q
 
dx = a sec2q dq
 
 
 
 
 
= log |sec q + tan q| + C1
 
 
 
 
 
 
 
 
 
Now, put so that dx = dt. Therefore, writing
 
08. To find integral of the type proceeding as in (7), we obtain the integral, using standard formulae.
 
09. To find the integral of the type where p, q, a, b, c are constants, we are to find real numbers A, B such that
 
 
To determine A and B, we equate from both sides the coefficients of x and the constant terms, A and B are thus obtained and hence, the integral is reduced to known forms.
 
10. For the evaluation of the integral of the type we proceed as above and transform the integral into known standard forms.
 
Let us illustrate the above methods by some examples.
 
Example:
 
Evaluate the following integral
 
The given integral is of the form (10). Using this formula, express
 
 
Equating the coefficients of x and the constant term
 
2A = 5 4A + B = 3
 
2(2A) + B =3
 
10 + B = 3
 
 
Therefore
 
 
 
 
Put x2 + 4x + 10 = t so that (2x + 4)dx=dt
 
 
 
 
 
 
 
 
Put x + 2 = t
 
dx = dt
 
 
 
 
Substituting (2) and (3) in (1), we have
 
 
 
 
 
  
     
   
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