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

MA 3972-MA-Book May 8, 2018 13:52

Integration 237

Step 3: Rewrite:

∫

√^1

9 −u^2

du

2

=

1

2

∫

√du

32 −u^2

.

Step 4: Integrate:

1

2

sin−^1

(

u

3

)

+C.

Step 5: Replaceu:

1

2

sin−^1

(

2 x

3

)

+C.

Step 6: Differentiate and Check:

1

2

1

√

1 −

(

2 x

3

) 2 ·

2

3

=

1

3

1

√

1 −

4 x^2

9

=

√^1

9

√^1

1 −

4 x^2

9

=

√^1

9

(

1 −

4 x^2

9

)

=

1

√

9 − 4 x^2

.

Example 2

Evaluate

∫

1

x^2 + 2 x+ 5

dx.

Step 1: Rewrite:

∫

1

(x^2 + 2 x+ 1 )+ 4

=

∫

1

(x+ 1 )^2 + 22

dx

=

∫

1

22 +(x+ 1 )^2

dx.

Letu=x+1.

Step 2: Differentiate:du=dx.

Step 3: Rewrite:

∫

1

22 +u^2

du.

Step 4: Integrate:

1

2

tan−^1

(

u

2

)

+C.

Step 5: Replaceu:

1

2

tan−^1

(

x+ 1

2

)

+C.

Step 6: Differentiate and Check:

(

1

2

) 1

(

1

2

)

1 +[(x+1)/ 2 ]^2

=

(

1

4

)

1

1 +(x+ 1 )^2 / 4

=

(

1

4

)

4

4 +(x+ 1 )^2

=

1

x^2 + 2 x+ 5

.

TIP • If the problem gives you that the diameter of a sphere is 6 and you are using formulas

such asv=

4

3

πr^3 ors= 4 πr^2 , do not forget thatr=3.