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Test 1 Review Problems: Indefinite and Improper Integrals - Prof. Allyson Alston, Exams of Calculus
The review problems for test 1 in ma 241-005 (spring 2006) focusing on evaluating indefinite and improper integrals. Students are required to solve problems involving basic integration techniques and improper integrals, along with approximating definite integrals using the midpoint rule, trapezoidal rule, and simpson’s rule.
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MA 241-005 (Spring 2006) Test 1 Review Problems Evaluate the following indefinite and indefinite integrals.
1. ! 5 cos( x )" 2 sec^2 ( x ) dx 2.! ( )
"
"
2 1 2
xx 4 dx 3.!
" 4 4 sin 2
x dx
4.!
" "
"
dx x x x 6
2 5.!
+
dx x x (^33) 2
6 .! sec^2 ( 2 x )tan( 2 x ) dx
7.! sin^ ( x^ )cos( x ) dx
3
8.! e^ x dx
2 x (^) sin( 3 )
9.! ( x^^ "^ ) e dx
1 x 1 0. dx x x x x x
! "
" + "
2
11.!
"
dx 7 x 2
12.!
"
dx x x (^21)
- dx x x e
!
1 ln
14.!
"
dx e e x x 1
15 .!
" xe dx 3 x^21
16.! x xdx
4
tan sec 17.!
2 0 3 xe dx x
18.!
" 0 x sin xdx Evaluate the following improper integrals. State whether the integral converges or diverges.
19.!
16 0 4 1
dx x
20.!
" 1
x ln xdx 21.!
"
1 0 1
dx x
- Use the Midpoint Rule, Trapezoidal Rule, and Simpson’s Rule with n = 4 to
approximate the definite integral!
+
2 0
dx x
.
Answers to Test 1 Practice Problems
- 5 sin x! 2 tan x + C
!
4.! 5 ln x! 3 + 6 ln x + 2 + C
5. ( )
1 (^2 3 ) 3 3 x + + C
tan ( 2 ) 4 x + C
4 sin 4 x
- C
8.^2
sin 3 cos 3 13 13 e x^ x! e x x + C
9.( x! 1 ) e x^! e x + C
2 x! 3 x + 4 ln x! 3 ln x! 1 + C
ln 7 2 7 x! + C
12.^2
ln 1 2 x! + C
14.ln e x! 1 + C
x e C !
2 3 tan tan 2 3 x x
- C
17.^6
e + 18.!
20.! " diverges 21.!" # diverges
- Midpoint: 1. Trapezoidal: 1. Simpson: 1.