Intergration

Question 1

What is the derivative of f(x) = x^n?

Answer 1

f’(x) = n * x^(n-1)

Question 2

True or False: The integral of a constant k with respect to x is kx + C.

Answer 2

True

Question 3

What is the integral of f(x) = x^3?

Answer 3

F(x) = (1/4)x^4 + C

Question 4

Fill in the blank: The derivative of sin(x) is __________.

Answer 4

cos(x)

Question 5

What is the power rule for integration?

Answer 5

∫x^n dx = (1/(n+1))x^(n+1) + C, for n ≠ -1

Question 6

What is the derivative of e^x?

Answer 6

e^x

Question 7

True or False: The second derivative of a function provides information about its concavity.

Answer 7

True

Question 8

What is the integral of cos(x)?

Answer 8

sin(x) + C

Question 9

What is the product rule for differentiation?

Answer 9

If u and v are functions, then (uv)’ = u’v + uv’.

Question 10

What is the integral of 1/x?

Answer 10

ln|x| + C

Question 11

Fill in the blank: The derivative of ln(x) is __________.

Answer 11

1/x

Question 12

What is the quotient rule for differentiation?

Answer 12

If u and v are functions, then (u/v)’ = (u’v - uv’)/v^2.

Question 13

What is the integral of sin(x)?

Answer 13

-cos(x) + C

Question 14

True or False: The integral of a function gives the area under the curve of that function.

Answer 14

True

Question 15

What is the derivative of tan(x)?

Answer 15

sec^2(x)

Question 16

What does C represent in the context of indefinite integrals?

Answer 16

The constant of integration

Question 17

Fill in the blank: The derivative of x^5 is __________.

Answer 17

5x^4

Question 18

What is the chain rule for differentiation?

Answer 18

If y = f(g(x)), then dy/dx = f’(g(x)) * g’(x).

Question 19

What is the integral of e^x?

Answer 19

e^x + C

Question 20

True or False: The Fundamental Theorem of Calculus connects differentiation and integration.

Answer 20

True

Question 21

What is the derivative of a constant k?

Answer 21

0

Question 22

Fill in the blank: The integral of x^2 is __________.

Answer 22

(1/3)x^3 + C

Question 23

What is the integral of sec^2(x)?

Answer 23

tan(x) + C

Question 24

What is the derivative of the function f(x) = 3x^2 + 2x + 1?

Answer 24

f’(x) = 6x + 2

Question 25

What is the integral of x^n for n = -1?

Answer 25

ln|x| + C

Question 26

True or False: Differentiation can be used to find the slope of a tangent line.

Answer 26

True

Question 27

What is the integral of (3x^2)?

Answer 27

x^3 + C

Question 28

What is the derivative of f(x) = cos(x)?

Answer 28

f'(x) = -sin(x)

Question 29

Fill in the blank: The derivative of a function at a point gives __________.

Answer 29

the slope of the tangent line at that point

Question 30

What is the integral of (2x)?

Answer 30

x^2 + C

Question 31

What is the derivative of f(x) = x^4 + 3x^2?

Answer 31

f'(x) = 4x^3 + 6x

Question 32

True or False: The integral can be used to find the total accumulation of a quantity.

Answer 32

True

Question 33

What is the integral of (sin(2x))?

Answer 33

-1/2 cos(2x) + C

Question 34

What is the derivative of f(x) = 5x^3 - 2x + 7?

Answer 34

f'(x) = 15x^2 - 2

Question 35

Fill in the blank: The integral of a function over an interval gives __________.

Answer 35

the net area under the curve

Question 36

What is the integral of (e^(3x))?

Answer 36

(1/3)e^(3x) + C

Question 37

What is the derivative of f(x) = 7sin(x)?

Answer 37

f'(x) = 7cos(x)

Question 38

True or False: The integral of a function can be used to find the average value of the function over an interval.

Answer 38

True

Question 39

What is the integral of (x^4)?

Answer 39

(1/5)x^5 + C

Question 40

What is the derivative of f(x) = 4/x?

Answer 40

f'(x) = -4/x^2

Question 41

Fill in the blank: The integral of (cos(3x)) is __________.

Answer 41

(1/3)sin(3x) + C

Question 42

What is the integral of (1/(x^2 + 1))?

Answer 42

arctan(x) + C

Question 43

What is the derivative of f(x) = 2^x?

Answer 43

f'(x) = 2^x ln(2)

Question 44

True or False: The product rule is used for finding the derivative of the product of two functions.

Answer 44

True

Question 45

What is the integral of (x^n)dx?

Answer 45

(1/(n+1))x^(n+1) + C, for n ≠ -1

Question 46

What is the derivative of f(x) = ln(x^2)?

Answer 46

f'(x) = 2/x

Question 47

Fill in the blank: The integral of (tan(x)) is __________.

Answer 47

-ln|cos(x)| + C

Question 48

What is the integral of (x^3 + 2x)?

Answer 48

(1/4)x^4 + x^2 + C