Practice Integration Z Math 120 Calculus I

Here¡¯s a list of practice exercises. There¡¯s a hint

for each one as well as an answer with intermediate

steps.

Practice Integration

Math 120 Calculus I

Z

1.

(x4 ? x3 + x2 ) dx. Hint. Answer.

D Joyce, Fall 2013

Z

(5t8 ? 2t4 + t + 3) dt. Hint. Answer.

This first set of indefinite integrals, that is, an- 2.

tiderivatives, only depends on a few principles of

Z

integration, the first being that integration is inverse to differentiation. Besides that, a few rules 3.

(7u3/2 + 2u1/2 ) du. Hint. Answer.

can be identified: a constant rule, a power rule,

linearity, and a limited few rules for trigonometric,

Z

logarithmic, and exponential functions.

4.

(3x?2 ? 4x?3 ) dx. Hint. Answer.

Z

k dx = kx + C, where k is a constant

Z

3

5.

dx. Hint. Answer.

Z

x

1

xn dx =

xn+1 + C, if n 6= ?1



n+1

Z 

7

4

Z

+

6.

dt. Hint. Answer.

1

3t2 2t

dx = ln |x| + C

x

Z

Z



Z 

3

¡Ì

kf (x) dx = k f (x) dx

7.

dy. Hint. Answer.

5 y?¡Ì

y

Z

Z

Z

(f (x) ¡À g(x)) dx = f (x) dx ¡À g(x) dx

Z

3x2 + 4x + 1

Z

8.

dx. Hint. Answer.

2x

sin x dx = ? cos x + C

Z

Z

9.

(2 sin ¦È + 3 cos ¦È) d¦È. Hint. Answer.

cos x dx = sin x + C

Z

ex dx = ex + C

Z

10.

(5ex ? e) dx. Hint. Answer.

Z

1

dx = arctan x + C

Z

1 + x2

4

Z

11.

dt. Hint. Answer.

2

1

1

+

t

¡Ì

dx = arcsin x + C

1 ? x2

Z

We¡¯ll add more rules later, but there are plenty here

12.

(ex+3 + ex?3 ) dx. Hint. Answer.

to get acquainted with.

Z

13.

1

¡Ì

7

du. Hint. Answer.

1 ? u2

Z 

14.

Z

15.

Z

16.

17.

1

r ? 2r +

r

2



Integrating polynomials is fairly easy, and you¡¯ll

get the hang of it after doing just a couple of them.

Answer.

dr. Hint. Answer.

4 sin x

dx. Hint. Answer.

3 tan x

Z

3. Hint.

You can use the power rule for other powers besides integers. For instance,

Z

u3/2 du = 52 u5/2 + C

(7 cos x + 4ex ) dx. Hint. Answer.

Z ¡Ì

3

(7u3/2 + 2u1/2 ) du.

7v dv. Hint. Answer.

Answer.

Z

18.

Z

19.

Z

20.

4

¡Ì dt. Hint. Answer.

5t

Z

4. Hint.

(3x?2 ? 4x?3 ) dx

You can even use the power rule for negative exponents (except ?1). For example,

Z

x?3 dx = ? 12 x?2 + C

1

dx. Hint. Answer.

3x2 + 3

¡Ì

x4 ? 6x3 + ex x

¡Ì

dx. Hint. Answer.

x

Answer.

Z

3

dx

x

This is 3x?1 and the general power rule doesn¡¯t

apply. But you can use

Z

1

dx = ln |x| + C.

x

5. Hint.

Z

1. Hint.

(x4 ? x3 + x2 ) dx.

Integrate each term using the power rule,

Z

1

xn dx =

xn+1 + C.

n+1

Answer.

So to integrate xn , increase the power by 1, then

6. Hint.

divide by the new power. Answer.

Z

2. Hint.



4

7

+

dt

3t2 2t

Treat the first term as 34 t?2 and the second term

as 72 t?1 . Answer.

(5t8 ? 2t4 + t + 3) dt.

Z 



Z 

Remember that the integral of a constant is the

3

¡Ì

5 y?¡Ì

dy

constant times the integral. Another way to say 7. Hint.

y

that is that you can pass a constant through the

It¡¯s usually easier to turn those square roots into

integral sign. For instance,

1

fractional powers. So, for instance, ¡Ì is y ?1/2 .

Z

Z

y

5t8 dt = 5 t8 dt

Answer.

2

Z

3x2 + 4x + 1

16. Hint.

(7 cos x + 4ex ) dx

dx

8. Hint.

2x

Just more practice with trig and exponential

Use some algebra to simplify the integrand, that

functions. Answer.

is, divide by 2x before integrating. Answer.

Z ¡Ì

Z

3

17. Hint.

7v dv

9. Hint.

(2 sin ¦È + 3 cos ¦È) d¦È

¡Ì

¡Ì

¡Ì

3

3

3

You can write

7v

as

7

v. And remember

Getting the ¡À signs right when integrating sines

¡Ì

1/3

3

you can write v as v . Answer.

and cosines takes practice. Answer.

Z

Z

4

¡Ì dt

18. Hint.

10. Hint.

(5ex ? e) dx

5t

Use algebra to write this in ¡Ì

a form that¡¯s easier to

Just as the derivative of ex is ex , so the integral

x

x

t is t?1/2 . Answer.

integrate.

Remember

that

1/

of e is e . Note that the ?e in the integrand is a

constant. Answer.

Z

1

dx

19. Hint.

Z

3x2 + 3

4

11. Hint.

dt

You can factor out a 3 from the denominator to

1 + t2

Remember that the derivative of arctan t is put it in a form you can integrate. Answer.

1

. Answer.

¡Ì

Z 4

1 + t2

x ? 6x3 + ex x

¡Ì

20. Hint.

dx

x

Z

¡Ì

Divide through by x before integrating. Alter12. Hint.

(ex+3 + ex?3 ) dx

natively, write the integrand as

When working with exponential functions, rex?1/2 (x4 ? 6x3 + ex x1/2 )

member to use the various rules of exponentiation. Here, the rules to use are ea+b = ea eb and

and multiply. Answer.

ea?b = ea /eb . Answer.

Z

Z

7

du

1 ? u2

Remember that the derivative of arcsin u is

Z

1

¡Ì

Answer.

1. Answer.

(x4 ? x3 + x2 ) dx.

1 ? u2

13. Hint.

¡Ì



Z 

The integral is 15 x5 ? 14 x4 + 13 x3 + C.

1

Whenever you¡¯re working with indefinite inte14. Hint.

r2 ? 2r +

dr

r

grals like this, be sure to write the +C. It signifies

Use the power rule, but don¡¯t forget the integral that you can add any constant to the antiderivative

of 1/r is ln |r| + C. Answer.

F (x) to get another one, F (x) + C.

When you¡¯re working

Z

Z bwith definite integrals with

4 sin x

15. Hint.

dx

limits of integration,

, the constant isn¡¯t needed

3 tan x

a

You¡¯ll need to use trig identities to simplify this. since you¡¯ll be evaluating an antiderivative F (x) at

Answer.

b and a to get a numerical answer F (b) ? F (a).

3

Z

8

Z

4

(5t ? 2t + t + 3) dt.

2. Answer.

10. Answer.

(5ex ? e) dx

That equals 5ex ? ex + C.

The integral is 95 t9 ? 52 t5 + 21 t2 + 3t + C.

Z

Z

3. Answer.

(7u

3/2

+ 2u

This integral evaluates as

Z

4. Answer.

1/2

4

dt.

1 + t2

That evaluates as 4 arctan t + C. Some people

prefer to write arctan t as tan?1 t.

11. Answer.

) du.

14 5/2

u

5

+ 43 u3/2 + C.

Z

(3x?2 ? 4x?3 ) dx.

12. Answer.

(ex+3 + ex?3 ) dx.

The integrand is its own antiderivative, that is,

That equals ?3x?1 +2x?2 +C. If you prefer, you

the integral is equal to

3

2

could write the answer as ? + 2 + C

x x

ex+3 + ex?3 + C.

Z

3

If you write the integrand as ex e3 + ex /e3 , and note

dx

5. Answer.

that e3 is just a constant, you can see that it¡¯s its

x

That¡¯s 3 ln |x|+C. The reason the absolute value own antiderivative.

sign is there is that when x is negative, the derivaZ

7

tive of ln |x| is 1/x, so by putting in the absolute 13. Answer.

¡Ì

du.

1 ? u2

value sign, you¡¯re covering that case, too.

The integral equals 7 arcsin u.



Z 



Z 

7

4

1

+

6. Answer.

dt.

2

2

14. Answer.

r ? 2r +

dr.

3t

2t

r

The integral of 43 t?2 + 72 t?1 is ? 34 t?1 + 27 ln |t| + C.

The integral evaluates as

Z 

7. Answer.

3

¡Ì

5 y?¡Ì

y

1 3

r

3



dy.

? r2 + ln |r| + C.

Z

4 sin x

15. Answer.

dx

The integral of 5y ?3y

is

3 tan x

10 ¡Ì

You could write that as 3 y y ? 6 y + C if you

The integrand simplifies to 43 cos x. Therefore the

prefer.

integral is 34 sin x + C.

Z

Z

3x2 + 4x + 1

8. Answer.

dx.

16. Answer.

(7 cos x + 4ex ) dx.

2x

The integral of 2x + 2 + 21 x?1 is

That¡¯s 7 sin x + 4ex + C.

1/2

?1/2

10 3/2

y ?6y 1/2 +C.

3 ¡Ì

Z ¡Ì

3

17. Answer.

7v dv.

1

x + 2x + ln |x| + C.

2

2

Since you can rewrite the integrand as

therefore its integral is

¡Ì 4/3

3 3

7 v + C.

4

Z

9. Answer.

(2 sin ¦È + 3 cos ¦È) d¦È.

That¡¯s equal to ?2 cos ¦È + 3 sin ¦È + C.

4

¡Ì

3

7 v 1/3 ,

Z

4

¡Ì dt.

5t

4 ?1/2

8

The integral of ¡Ì t

is equal to ¡Ì t1/2 + C.

5

5

p

You could also write that as 8 t/5 + C.

18. Answer.

Z

1

dx

+3

This integral equals 13 arctan x + C.

19. Answer.

3x2

¡Ì

x4 ? 6x3 + ex x

¡Ì

20. Answer.

dx.

x

The integral can be rewritten as

Z

(x7/2 ? 6x5/2 + ex ) dx

Z

which equals 92 x9/2 ?

12 7/2

x

7

+ ex + C.

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