Improper Integrals

Question 1

Evaluate the integral from 1 to infinity of 1/x^2 dx.

Answer 1

1 (Convergent).

Question 2

Evaluate the integral from 1 to infinity of 1/x dx.

Answer 2

Divergent (Natural log approaches infinity).

Question 3

Evaluate the integral from 0 to 1 of 1/sqrt(x) dx.

Answer 3

2 (Convergent Type II integral).

Question 4

Evaluate the integral from -infinity to 0 of e^x dx.

Answer 4

1 (Convergent).

Question 5

Evaluate the integral from 0 to infinity of 1/(1+x^2) dx.

Answer 5

pi/2 (arctan(infinity) - arctan(0)).

Question 6

Does the integral from 1 to infinity of 1/x^p dx converge if p = 1.001?

Answer 6

Yes (Converges for all p > 1).

Question 7

Does the integral from 1 to infinity of 1/x^p dx converge if p = 0.99?

Answer 7

No (Diverges for all p <= 1).

Question 8

Evaluate the integral from 0 to 8 of x^(-1/3) dx.

Answer 8

6 (Convergent).

Question 9

Evaluate the integral from 0 to infinity of e^(-2x) dx.

Answer 9

1/2 (Convergent).

Question 10

Evaluate the integral from 2 to infinity of 1/(x*ln(x)) dx.

Answer 10

Divergent (Let u = ln x, then integrate 1/u).

Question 11

Evaluate the integral from 0 to 2 of 1/(x-1) dx.

Answer 11

Divergent (Infinite discontinuity at x = 1).

Question 12

Evaluate the integral from 0 to 1 of ln(x) dx.

Answer 12

-1 (Use integration by parts and limits).

Question 13

Evaluate the integral from -infinity to infinity of 1/(1+x^2) dx.

Answer 13

pi (Split into two integrals, each is pi/2).

Question 14

Evaluate the integral from 1 to infinity of ln(x)/x dx.

Answer 14

Divergent (Use u-substitution u = ln x).

Question 15

Does the integral from 1 to infinity of sin^2(x)/x^2 dx converge?

Answer 15

Yes (By Comparison Test with 1/x^2).

Question 16

Evaluate the integral from 0 to infinity of x*e^-x dx.

Answer 16

1 (Use integration by parts).

Question 17

Evaluate the integral from 0 to pi/2 of tan(x) dx.

Answer 17

Divergent (Vertical asymptote at pi/2).

Question 18

Evaluate the integral from e to infinity of 1/(x*(ln x)^2) dx.

Answer 18

1 (Let u = ln x).

Question 19

Evaluate the integral from -1 to 1 of 1/x^2 dx.

Answer 19

Divergent (Infinite discontinuity at x = 0).

Question 20

Evaluate the integral from 0 to infinity of x/(1+x^2) dx.

Answer 20

Divergent (Natural log approaches infinity).

Question 21

Evaluate the integral from 1 to infinity of 1/(x^2 + x) dx.

Answer 21

ln(2) (Use partial fractions).

Question 22

Does the integral from 1 to infinity of 1/sqrt(x^3+1) dx converge?

Answer 22

Yes (Compare with x^-1.5).

Question 23

Evaluate the integral from 0 to 4 of 1/sqrt(16-x^2) dx.

Answer 23

pi/2 (arcsin(x/4) from 0 to 4).

Question 24

Evaluate the integral from -infinity to infinity of x*e^(-x^2) dx.

Answer 24

0 (Odd function over a symmetric infinite interval).

Question 25

Evaluate the integral from 0 to 1 of e^x/(e^x - 1) dx.

Answer 25

Divergent (Infinite discontinuity at x = 0).

Question 26

Evaluate the integral from 2 to infinity of 2/(x^2 - 1) dx.

Answer 26

ln(3) (Use partial fractions).

Question 27

Does the integral from 0 to infinity of cos(x) dx converge?

Answer 27

No (Limit of sin(b) as b goes to infinity does not exist).

Question 28

Evaluate the integral from 0 to 1 of 1/x^0.5 dx.

Answer 28

2 (Converges because p < 1 for Type II).

Question 29

Evaluate the integral from 1 to infinity of e^-sqrt(x)/sqrt(x) dx.

Answer 29

2/e (Let u = sqrt(x)).

Question 30

Evaluate the integral from 0 to infinity of 1/x^2 dx.

Answer 30

Divergent (Fails at both x=0 and infinity).