Integrals

Question 1

integral of cos(x)dx =

Answer 1

sin(x) + c

Question 2

integral of sin(x)dx =

Answer 2

-cos(x) +c

Question 3

∫a dx =

Answer 3

ax + C

Question 4

∫x dx

Answer 4

x2/2 + C

Question 5

∫x2 dx

Answer 5

x3/3 + C

Question 6

∫(1/x) dx

Answer 6

ln|x| + C

Question 7

∫ex dx

Answer 7

ex + C

Question 8

∫ax dx

Answer 8

ax/ln(a) + C

Question 9

∫ln(x) dx

Answer 9

x ln(x) − x + C

Question 10

∫cos(x) dx

Answer 10

sin(x) + C

Question 11

∫sin(x) dx

Answer 11

-cos(x) + C

Question 12

∫sec2(x) dx

Answer 12

tan(x) + C

Question 13

∫cf(x) dx

Answer 13

c∫f(x) dx

Question 14

∫x^n dx

Answer 14

(x^n+1)/(n+1) + C

Question 15

∫(f + g) dx

Answer 15

∫f dx + ∫g dx

Question 16

∫(f - g) dx

Answer 16

∫f dx - ∫g dx

Question 17

When doing integration by parts, a good rule of thumb is to …

Answer 17

choose a u that gets simpler when you differentiate it and a v that doesn’t get any more complicated when you integrate it.

Question 18

A helpful rule of thumb when integrating by parts is ______. Choose u based on which of these comes first:

Answer 18

I LATE

I: Inverse trigonometric functions such as sin^-1(x), cos^-1(x), tan^-1(x)
L: Logarithmic functions such as ln(x), log(x)
A: Algebraic functions such as x2, x3
T: Trigonometric functions such as sin(x), cos(x), tan (x)
E: Exponential functions such as ex, 3x

Question 19

sin^2(x) + cos^2(x) =

Answer 19

1

Question 20

Partial Fraction decomposition only works when _______________, and if it’s not in this form, we need to _______________ first.

Answer 20

the degree of the top is less than the bottom.

do polynomial long division first.

Question 21

In partial fraction decomposition, we reduce until we have complex numbers.

Answer 21

FALSE

we reduce until we get an irreducible form (a fraction that if further reduced breaks into complex numbers)

Question 22

In partial fraction decomposition, When you have a quadratic factor you need to include this partial fraction:

Answer 22

(B1x + C1) / (Your Quadratic)

Question 23

In partial fraction decomposition, sometimes you may get a factor with an exponent, like (x−2)^3. When this happens, you …

The same thing can also happen to quadratics:
1/(x2+2x+3)^2 has the partial fractions …

Answer 23

need a partial fraction for each exponent from 1 up.

A1/(x−2) + A2/(x−2)^2 + A3/(x−2)^3

(B1x + C1)/(x2+2x+3) + (B2x + C2)/(x2+2x+3)^2

Question 24

In partial fraction decomposition, even after using the roots (zeros) of the bottom you can end up with unknown constants.

So the next thing to do is:

Answer 24

Gather all powers of x together and then solve it as a system of linear equations.