MAT 21B Lecture Notes - Lecture 22: Trigonometric Substitution, Additive Inverse, Additive Identity
MAT 21B – Lecture 22 – Trigonometric Function Substitution
• This method of integration is good if you are out of ideas for trying u-substitution
and integration by parts AND integral that contains either of the following in
some form regardless of any change in the numbers:
i. , then try because
ii. , then try because
iii. , then try because
• Recall that has a domain of and does not have an inverse.
• The additive inverse of is because
Additive identity is 0 because if we have a number a, then a + 0 = a.
• The multiplicative inverse of is because
Multiplicative identity is 1 because a * 1 = a.
• The (compositional) functional inverse of is because
Recall that . If and
, then . Functional identity is x
because .
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MAT 21B Full Course Notes
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Additive identity is 0 because if we have a number a, then a + 0 = a. (cid:2869)si(cid:2924)(cid:4666)(cid:4667)=(cid:883). (cid:3030)(cid:3042)(cid:3046)(cid:3118)(cid:4666)(cid:4667)(cid:1856)= c(cid:2925)s(cid:4666)(cid:4667) (cid:2873) (cid:2871) (cid:2872)(cid:3118)(cid:1856)(cid:1876: notation: [sin (cid:4666)(cid:1876)(cid:4667)] (cid:2869)= (cid:2869)si(cid:2924) (cid:4666)(cid:4667)=csc (cid:4666)(cid:1876)(cid:4667); however, (cid:1871)(cid:1866) (cid:2869)(cid:4666)(cid:1876)(cid:4667)=arcsin (cid:4666)(cid:1876)(cid:4667), example: use trigonometric substitution to evaluate (cid:2869) (cid:2869) (cid:3118)(cid:1856)(cid:1876) Then (cid:2869) solving for , we obtain (cid:1876)=sin(cid:4666)(cid:4667)(cid:3643)arcsin(cid:4666)(cid:1876)(cid:4667)=. Thus, (cid:883)(cid:1856)=+= arcsin(cid:4666)(cid:1876)(cid:4667)+: example (u-substitution): if (cid:4666)(cid:1876)(cid:4667)= (cid:1871)(cid:1866)(cid:2870)(cid:4666)(cid:1876)(cid:4667)cos (cid:4666)(cid:1876)(cid:4667)(cid:1856)(cid:1876) and (cid:1873)=sin(cid:4666)(cid:1876)(cid:4667)(cid:3642) (cid:1856)(cid:1873)=cos(cid:4666)(cid:1876)(cid:4667)(cid:1856)(cid:1876), then (cid:4666)(cid:1876)(cid:4667)= (cid:1873)(cid:2870) (cid:1856)(cid:1873)=(cid:2869)(cid:2871)(cid:1873)(cid:2871)+=(cid:2869)(cid:2871)(cid:1871)(cid:1866)(cid:2871)(cid:4666)(cid:1876)(cid:4667)+. substitution, the old variable (often x) is a function of a new variable, while in, example 2: compute (cid:2873) (cid:2871) (cid:2872)(cid:3118)(cid:1856)(cid:1876) and (cid:1876)= (cid:2871)(cid:2872)sin(cid:4666)(cid:4667)(cid:3642)(cid:1856)(cid:1876)= (cid:2871)(cid:2872)cos(cid:4666)(cid:4667)(cid:1856). (cid:2869) (cid:3046)(cid:3041)(cid:3118)(cid:4666)(cid:4667)(cid:1856)= (cid:2873) u-substitution, the new variable, u is a function of the old variable (often x): these substitution methods are essentially reverse processes of each other. (cid:2871) (cid:2871)(cid:3046)(cid:3041)(cid:3118)(cid:4666)(cid:4667)(cid:1856)= (cid:2873) (cid:2871)(cid:1499) (cid:2871)(cid:2870) c(cid:2925)s(cid:4666)(cid:4667: difference between trigonometric and u-substitutions: in trigonometric. (cid:2871)(cid:2872)sin(cid:4666)(cid:4667)(cid:3643) (cid:2870) (cid:2871)(cid:1876)=sin(cid:4666)(cid:4667)(cid:3643)=arcsin(cid:4672)(cid:2870) (cid:2871)(cid:1876)(cid:4673). represented by (cid:1876)(cid:2870)+(cid:1877)(cid:2870)=(cid:883) from x = t to x = 1 such that (cid:883) (cid:1872) (cid:883).
