240 STEP 4. Review the Knowledge You Need to Score High
Letu=x^2 ; then
du
dx
= 2 x.Rewrite:y=−∫u1sintdt.dy
dx=
dy
du·
du
dx
=(−sinu)2x=(−sinx^2 )2x=− 2 xsin(x^2 )Example 5Find
dy
dx
;ify=∫x 2x√
et+ 1 dt.y=∫ 0x√
et+ 1 dt+∫x 20√
et+ 1 dt=−∫x0√
et+ 1 dt+∫x 20√
et+ 1 dt=
∫x 20√
et+ 1 dt−∫x0√
et+ 1 dtSincey=∫ x 20√
et+ 1 dt−∫ x0√
et+ 1 dtdy
dx=
(
d
dx∫x 20√
et+ 1 dt)
−(
d
dx∫x0√
et+ 1 dt)=
(√
ex^2 + 1)
d
dx
(x^2 )−(√
ex+ 1)= 2 x√
ex^2 + 1 −√
ex+ 1.Example 6F(x)=∫x1(t^2 −4)dt, integrate to findF(x) and then differentiate to findf′(x).F(x)=
t^3
3
− 4 t]x
1 =(
x^3
3
− 4 x)
−(
13
3−4(1)
)=
x^3
3
− 4 x+11
3
F′(x)= 3(
x^2
3)
− 4 =x^2 − 4