So ∫
sinxcosxdx=1
2∫
sin 2xdx=−1
4cos 2x+CRefer back to Review Question 9.1.6(iv) for an alternative form of
this.9.2.9 Using trig substitutions in integration
➤
252 283➤Trig or hyperbolic substitutions may help you to deal with some rational and irrational
functions. Thus if you have
√
a^2 −x^2 tryx=asinθ orx=tanhθ
√
x^2 −a^2 tryx=acoshθ orx=secθ
√
a^2 +x^2 tryx=asinhθ orx=tanθThe form of the substitution may depend on the range of values ofx.
Another useful trig substitution is
t=tanθ/ 2With this we have
cosθ=1 −t^2
1 +t^2, sinθ=2 t
1 +t^2We also have to change betweendθanddtof course. We have
dθ
dt=1
2sec^2θ
2=1
2(
1 +tan^2θ
2)
=1
2( 1 +t^2 )from which
dθ=2 dt
1 +t^2This substitution can sometimes be used to convert an integral of the form
∫
f(cosθ,sinθ)dθto an integral of a rational function int, which may be easier to deal with.