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