![]() Normally you can see by differentiation that the solution that is found is valid also for #x in (-oo, -a)#Ĭ. #int f(x)dx = int R(asect, atant)sect tant dt# Restrict the function to #x in (a,+oo)# and substitute: #x = asect#, #dx = asect tantdt# with #t in (0,pi/2)# and use the trigonometric identity:Ĭonsidering that for #t in (0,pi/2)# the tangent is positive: Here’s a table summarizing thesubstitution to make in each of the three kinds. The substitution willsimplify the integrand since it will eliminate the square root. Let #f(x)# be a rational function of #x# and #sqrt(x^2-a^2)#: In eachkind you substitute forxa certain trig function of new variable. #int f(x)dx = int R(atant, asect)sec^2t dt#ī. Substitute: #x = atant#, #dx = asec^2tdt# with #t in (-pi/2,pi/2)# and use the trigonometric identity:Ĭonsidering that for #t in (-pi/2,pi/2)# the secant is positive: Let #f(x)# be a rational function of #x# and #sqrt(x^2+a^2)#: In general trigonometric substitutions are useful to solve the integrals of algebraic functions containing radicals in the form #sqrt(x^2+-a^2)# or #sqrt(a^2+-x^2)#. Joe Foster Common Trig Substitutions: The following is a summary of when to use each trig substitution.
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