5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 15:57

Def inite Integrals 255

Example 1

Evaluate

∫π

π/ 4

cos(2t)dt.

Letu= 2 t;du= 2 dtor
du
2

=dt.

∫

cos(2t)dt=

∫

cosu

du

2

=

1

2

∫

cosudu

=

1

2

sinu+C=

1

2

sin(2t)+C

∫x

π/ 4

cos(2t)dt=

1

2

sin( 2 t)

]x

π/ 4

=

1

2

sin(2x)−

1

2

sin

(

2

(

π

4

))

=

1

2

sin(2x)−

1

2

sin

(

π

2

)

=

1

2

sin(2x)−

1

2

Example 2

Ifh(x)=

∫ x

3

√

t+ 1 dt, findh′(8).

h′(x)=

√

x+1;h′(8)=

√

8 + 1 = 3

Example 3

Find
dy
dx

;ify=

∫ 2 x

1

1

t^3

dt.

Letu= 2 x; then
dy
dx

= 2.

Rewrite:y=

∫u

1

1

t^3

dt.

dy

dx

=

dy

du

·

du

dx

=

1

u^3

·(2)=

1

(2x)^3

· 2 =

1

4 x^3

Example 4

Find
dy
dx

;ify=

∫ 1

x^2

sintdt.

Rewrite:y=−

∫x 2

1

sintdt.