t ) dt Thus I = ( ( )) f g t g t dt ′ This change of variable formula is one of the important tools available to us in the name of integration by substitution. It is often important to guess what will be the useful substitution. Usually, we make a substitution for a function whose derivative also occurs in the integrand as illustrated in the following examples. Example Integrate the following functions w.r.t.
x : sin mx x sin ( x + ) (iii) (iv) sin (tan – x Solution We know that derivative of mx is m . Thus, we make the substitution mx = t so that mdx = dt . mx dx t dt m = – m cos t + C = – m cos mx + C Derivative of x + is x . Thus, we use the substitution x + = t so that x dx = dt .
sin ( t dt = – cos t + C = – cos ( x + ) + C (iii) Derivative of x is . Thus, we use the substitution so that giving dx = t dt . Thus, tan t dt tan t dt Again, we make another substitution tan t = u so that sec t dt = du INTEGRALS tan t dt u du u + tan t + (since u = tan t ) tan C (since Hence, tan x + Alternatively , make the substitution tan (iv) Derivative of – x . Thus, we use the substitution tan – x = t so that = dt.
Therefore , sin (tan – x dx t dt = – cos t + C = – cos(tan – x ) + C Now, we discuss some important integrals involving trigonometric functions and their standard integrals using substitution technique. These will be used later without reference.