Basic Calculus

Question 1

What is the formula for integration by parts?

Answer 1

Question 2

Answer 2

= -cos(x) + C

Question 3

Answer 3

= tan(x) + C

Question 4

Answer 4

= -cot(x) + C

Question 5

Answer 5

= sin(x) + C

Question 6

Answer 6

= sec(x) + C

Question 7

Answer 7

= -csc(x) + C

Question 8

What is the substitution rule for definite integrals?

Answer 8

If g’ is continuous on [a,b] and if f is continuous on the range of u=g(x) then:

Question 9

What is the basic substitution rule, not necessarily for definite integrals?

Answer 9

If u=g(x) is a differentiable function whose range is an interval I, and f is continuous on I, then

Question 10

if the area is bounded by curves y=f(x) and y=g(x), and the lines x=a and x=b, where f and g are continuous and f(x) > g(x) for all x in [a,b];

What is the formula for the area between the two curves?

Answer 10

Question 11

If the function is symmetric and f is continuous on

[-a,a] and f is even such that [f(-x)=f(x)] then

Answer 11

Question 12

If the function is symmetric and f is continuous on

[-a,a] and f is odd such that [f(-x)=-f(x)] then

Answer 12

= 0

Question 13

The fundamental theorem of calculus states that if f is continuous on [a,b], then:

Answer 13

= F(b) - F(a)

Where F is any antiderivative of f. That is, a function such that F’=f.

Question 14

Using the basic properties of integrals, and assuming that f and g are continuous:

Answer 14

= c (b-a)

Question 15

Using the basic properties of integrals, and assuming that f and g are continuous:

Answer 15

Question 16

Using the basic properties of integrals, and assuming that f and g are continuous:

Answer 16

Question 17

Using the basic properties of integrals, and assuming that f and g are continuous:

Answer 17

Question 18

Using the properties of definite integrals:

Answer 18

Question 19

Using the properties of definite integrals:

Answer 19

= 0

Question 20

Using the properties of definite integrals:

Answer 20

Question 21

Using sigma notation rules:

Answer 21

= nc

Question 22

Using sigma notation rules:

Answer 22

Question 23

Using sigma notation rules:

Answer 23

Question 24

Using sigma notation rules:

Answer 24

Question 25

The net change theorem states that the rate of change

Answer 25

= F(b) - F(a)

the net change

Question 26

Using the properties of basic indefinite integrals;

Answer 26

Question 27

Using the properties of basic indefinite integrals:

Answer 27

Question 28

Using the properties of basic indefinite integrals:

Answer 28

= kx + C

Question 29

Using the properties of basic indefinite integrals:

Answer 29