Definite Integrals

Question 1

Evaluate the integral of f(x) from a to a.

Answer 1

0 (The area under a single point is zero).

Question 2

If the integral from 1 to 5 of f(x) is 10, what is the integral from 5 to 1?

Answer 2

-10 (Reversing the limits of integration negates the value).

Question 3

Simplify: Integral from 0 to 2 plus Integral from 2 to 7.

Answer 3

Integral from 0 to 7 of f(x) dx (Additive property).

Question 4

If Int(0 to 3) f=4 and Int(0 to 3) g=-2, find Int(0 to 3) of [2f + g].

Answer 4

2(4) + (-2) = 6.

Question 5

Is the integral from -2 to 2 of x^3 equal to 0? Why?

Answer 5

Yes. x^3 is an odd function and the interval is symmetric about the origin.

Question 6

Evaluate the integral from 0 to 3 of (x^2 + 1) dx.

Answer 6

12 (Calculated as [x^3/3 + x] from 0 to 3).

Question 7

Evaluate the integral from 1 to 4 of 1/sqrt(x) dx.

Answer 7

2 (Calculated as [2*sqrt(x)] from 1 to 4).

Question 8

Evaluate the integral from 0 to pi of sin(x) dx.

Answer 8

2 (Calculated as [-cos(x)] from 0 to pi).

Question 9

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

Answer 9

e - 1 (Calculated as [e^x] from 0 to 1).

Question 10

Solve the integral from 1 to 2 of 3/x dx.

Answer 10

3 ln(2) (Calculated as [3*ln|x|] from 1 to 2).

Question 11

FTC1: Find the derivative with respect to x of the integral from 1 to x of cos(t^2) dt.

Answer 11

cos(x^2).

Question 12

FTC1: Find the derivative with respect to x of the integral from 0 to x^2 of sin(t) dt.

Answer 12

2x*sin(x^2) (Apply the Chain Rule).

Question 13

Evaluate the integral from 0 to 2 of x*e^(x^2) dx using u-sub.

Answer 13

1/2(e^4 - 1).

Question 14

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

Answer 14

1/2 (Let u = ln(x)).

Question 15

Evaluate the integral from 0 to pi/4 of sec^2(x) dx.

Answer 15

1 (Calculated as [tan(x)] from 0 to pi/4).

Question 16

Find the average value of f(x) = x^2 on [0, 3].

Answer 16

3 (1/(3-0) * Integral from 0 to 3 of x^2).

Question 17

If f(x) is even and the integral from 0 to 5 is 7, what is the integral from -5 to 5?

Answer 17

14 (Symmetric property of even functions).

Question 18

Solve the integral from 0 to 1 of (4x^3 - 6x^2) dx.

Answer 18

-1 (Calculated as [x^4 - 2x^3] from 0 to 1).

Question 19

Find the derivative with respect to x of the integral from x to 5 of sqrt(t^3 + 1) dt.

Answer 19

-sqrt(x^3 + 1) (Negative sign due to x being the lower limit).

Question 20

Evaluate the integral from 0 to 3 of |x - 1| dx.

Answer 20

2.5 (Split into two triangles or two integrals at x=1).

Question 21

Evaluate the integral from 0 to pi/2 of cos(x)*e^sin(x) dx.

Answer 21

e - 1 (Let u = sin(x)).

Question 22

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

Answer 22

2 (Calculated as [x^2/2 - 1/x] from 1 to 2).

Question 23

Find the derivative with respect to x of the integral from 2 to sin(x) of t^2 dt.

Answer 23

sin^2(x)*cos(x) (Apply the Chain Rule).

Question 24

Evaluate the integral from 0 to 4 of sqrt(x) dx.

Answer 24

16/3 (Calculated as [2/3 * x^(3/2)] from 0 to 4).

Question 25

Solve the integral from -1 to 1 of 5 dx.

Answer 25

10 (Area of a rectangle with height 5 and width 2).

Question 26

Evaluate the integral from 0 to 1 of x(x^2+1)^3 dx.

Answer 26

15/8 (Let u = x^2 + 1).

Question 27

If Int(0 to 10) f=15 and Int(4 to 10) f=7, find Int(0 to 4) f.

Answer 27

8 (Using the additive property of intervals).

Question 28

Evaluate the integral from 0 to pi of (2 + sin(x)) dx.

Answer 28

2*pi + 2 (Calculated as [2x - cos(x)] from 0 to pi).

Question 29

Find the derivative of g(x) = integral from 1 to x^3 of 1/t dt.

Answer 29

3/x (Using FTC1 and the Chain Rule).

Question 30

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

Answer 30

pi/4 (Calculated as [arctan(x)] from 0 to 1).