Some Special Types of Integrals

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Following are few special integrals which can be integrated by using integration by parts

(i) Prove that

Proof:

Taking 1 as the second function and integrating by parts, we have

(ii). Prove that

Proof:

Taking 1 as the second function and integrating by parts, we have

(iii) Prove that

Proof:

Taking 1 as the second function

Integral of the form

Method:

The quadratic expression ax² + bx + c can be expressed in the form a(x²  ± A²) by the method of completing the square. The integrals can be evaluated by using the special integrals.

Integrals of the form

Method:

(ii) Find the values of the constants L and M by comparing the co-efficients of like powers of x on both sides.

(iv) The integrals in the form are easily integrable.

Example:

Suggested answer:

Put x+1 = t

dx = dt

(using formula (iii))