Mastering Mathematics Smartly
by Wee Wen Shih

A typical problem usually involves these steps:

(1) Carry out differentiation on the substitution, with respect to the original variable.

(2) Find the new lower and upper limit values, based on the substitution.

(3) Apply the substitution on the original expression.

(4) Use chain rule, to evaluate the integral with respect to the new variable.

The resulting expression should then be easy to integrate directly. At times, you may need to use trigonometric identities for integration.

See example.

An extension: Suppose that, for the above above, we exclude the lower and upper limits so that we have an indefinite integral. We will need to convert the expressions involving x to their equivalents involving u. This will require us to make use of the substitution:

• The first term 1/16 x becomes 1/16 inverse cosine 2u.
  
  
• For the second term, some work will be involved to use a double-angle trigonometric identity twice:
  1/64 sin 4x
  = (1/64) . 2sin 2x . cos 2x
  = (1/32) . (2 sin x . cos x) . (2 cos^2 x - 1) --- (*)
  
  Using a right-angled triangle (draw it!) and the fact that cos x = 2u, we are able to obtain sin x = sqrt(1 - 4u^2). The expression of sin x as well as the expresion of cos x are then substituted into the above expression indicated with asterisk, i.e. (*).

Try this! Try it again if the integral is an indefinite one.