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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:





04.
Prove that

Proof:

dx = a secq tanq dq




05.
Prove that

Proof:



q + C

06.
Prove that

Proof:

dx = a sec2q dq






Now, put
so that dx = dt. Therefore, writing

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 

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





