MATH 181 Final Exam Answer Key

MATH 181
Final exam
Answer key




1. Compute the definite integral:
Z π
x cos(2x)dx
0

Answer: 0.



2. Find the following indefinite integrals:


√
Z
x 2
√ dx = (x − 2)3/2 + 4 x − 2 + C
x−2 3
Z
1 1
x3 sin(x2 )dx = − x2 cos(x2 ) + sin(x2 ) + C
2 2
x−2
Z
dx 1
2
= ln +C
x +x−6 5 x+3
Z
dx 2 2
2
= √ tan−1 ( √ (x + 1/2)) + C
x +x+3 11 11
Z
dx 1 1
3
= − ln |x| + ln |x − 1| + ln |x + 1| + C
x −x 2 2
Z 7 7
x x
x6 ln xdx = ln x − +C
7 49
Z
1
arctan xdx = x tan−1 x − ln(1 + x2 ) + C
2
√ √ √ √
Z
cos( x)dx = 2 x sin x + cos x + C
Z
1 1 1
x2 e2x dx = x2 e2x − xe2x + e2x + C
2 2 4

1