Integration by substitution:

Direct Substitution

If integral is of the form ∫f(g(x)) g'(x) dx, then put g(x) = t, given that ∫f(t) exists.

(i).  = ln |f (x)| + c        

            Put f (x) = t => f' (x) dx = dt =>  = ln |t| + c = ln |f (x)| + c.

(ii).      

Indirect Substitution

If the integrand is of form f(x)g(x), where  g(x) is a function of the integral of f(x), then put integral of f(x) = t.

Example: Evaluate .

Solution:        Let lnx = t. Then dt = 1/x dx

                        Hence I = ∫sint dt = -cost + c = -cos(lnx) + c

Example: Evaluate.    

Solution: 

Example: Evaluate .

Solution:        Let  z = 2x³ + 3x

                        dz=  (6x² +3)dx = 3( 2x² +1)dx

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