45 Substitution
*Any Jervative rule
can be reversed to nd com untideminative mul
Example (4x³ cos(x²) de
ucx² de = 4x³lx
#(sin(x)) = 4x³ cos(x²)
so sex?cor (x8) lx = vin (x+4)+(
Lets formalize
hot Fas le on antiderivative of fax)
✓ ( Fig(x) = F'(g(a)) '(k) = f(g(x)) g`as
{f(g(x))g(x)dx= Fig(x)) +tc
let uagat)
= F(~)+(
= Stensdy
Theven: If u= g(x? is differentiable and I is continuous, then
Sagrgaslide = sher
fi fi
,45 Cont
a) x ( x
u = x² 4 Ju = 2x dx.
½
4696
12. 1/4 4 164
3342
3
+C
1647
+(
6) Stan (4) secies dr
3
Su³ du
安
+C
Herste
ždux day
uxton(x)
J = 0 (x)
с. Прогли
for 5 bi
{S Ju du
ไร
³ (3 + 2)+c
15 + 2) = (
2(sec (4)]+[
21(5)+(
ال
2 sec(u)tacus de
dux
=
2=5x
*Any Jervative rule
can be reversed to nd com untideminative mul
Example (4x³ cos(x²) de
ucx² de = 4x³lx
#(sin(x)) = 4x³ cos(x²)
so sex?cor (x8) lx = vin (x+4)+(
Lets formalize
hot Fas le on antiderivative of fax)
✓ ( Fig(x) = F'(g(a)) '(k) = f(g(x)) g`as
{f(g(x))g(x)dx= Fig(x)) +tc
let uagat)
= F(~)+(
= Stensdy
Theven: If u= g(x? is differentiable and I is continuous, then
Sagrgaslide = sher
fi fi
,45 Cont
a) x ( x
u = x² 4 Ju = 2x dx.
½
4696
12. 1/4 4 164
3342
3
+C
1647
+(
6) Stan (4) secies dr
3
Su³ du
安
+C
Herste
ždux day
uxton(x)
J = 0 (x)
с. Прогли
for 5 bi
{S Ju du
ไร
³ (3 + 2)+c
15 + 2) = (
2(sec (4)]+[
21(5)+(
ال
2 sec(u)tacus de
dux
=
2=5x
