1.4 Method of Substitution
i
the ofthe chain rule
analog for integrals
recall:f(g(x))'
f11g(x))g((x) =
So: (f'(g(x))g'(x)dx f(g(x)) c =
+
steps for substitution:
① Se u g(x) +
=
② compute du, du g'(x) dx
=
③ substitute
u and du,
(f'(g(x)) g'(x) (f'cu)du =
④ solve the
integral, Sfculdu"=fay C
+
⑤ substitute the original variable back in , f(g(x)) + C
examples:
xz
1) S2xe dx
a) u xzdu 2xdx
= =
Sixexx Se"du =
eu C
= +
(x +C
=
b) u ex-
=
eX" 2xdx du =
Sixe*dx=Sdu
u
=
C
+
-
ex
=
C
+
2) Sxexdx
u =
=txdx ydu
x-du 2xdx =
=
Sex(xdx):Se".Adu
:kSe"du
=key C +
=(ex+C
3)
Sxdy
u 1
=
-
xzdu =
- 2xdX
Sizudu Ifudu
f 31
xdx
=
=
-
x2 =
((2)(3h)u=3 C
+
3/y(1 x2)*3 C
-
= -
+
4)
Sindyday
u =
du 1xdx
mx =
VzSudu
Jinxdx S/zudu =
=
=
(z u-z C +
(642 C
=
+
(((nX)"
=
C
+
i
the ofthe chain rule
analog for integrals
recall:f(g(x))'
f11g(x))g((x) =
So: (f'(g(x))g'(x)dx f(g(x)) c =
+
steps for substitution:
① Se u g(x) +
=
② compute du, du g'(x) dx
=
③ substitute
u and du,
(f'(g(x)) g'(x) (f'cu)du =
④ solve the
integral, Sfculdu"=fay C
+
⑤ substitute the original variable back in , f(g(x)) + C
examples:
xz
1) S2xe dx
a) u xzdu 2xdx
= =
Sixexx Se"du =
eu C
= +
(x +C
=
b) u ex-
=
eX" 2xdx du =
Sixe*dx=Sdu
u
=
C
+
-
ex
=
C
+
2) Sxexdx
u =
=txdx ydu
x-du 2xdx =
=
Sex(xdx):Se".Adu
:kSe"du
=key C +
=(ex+C
3)
Sxdy
u 1
=
-
xzdu =
- 2xdX
Sizudu Ifudu
f 31
xdx
=
=
-
x2 =
((2)(3h)u=3 C
+
3/y(1 x2)*3 C
-
= -
+
4)
Sindyday
u =
du 1xdx
mx =
VzSudu
Jinxdx S/zudu =
=
=
(z u-z C +
(642 C
=
+
(((nX)"
=
C
+
