Integral ToolKit

Question 1

What is the most Important thing to never forget when answering an Integral?

Answer 1

+ C

Question 2

In a Trig Integral what has to happen if there is an inner function?

Answer 2

The answer must be divided by the derivative of the inner function.

Question 3

What alternative method should be used if a fraction can not be split into partial fractions?

Answer 3

U-Substitution

Question 4

How would you integrate 3/x^2?

Answer 4

Using Normal Integration.

= -3/x

Question 5

When using U-substitution how does one determine how the dx must change?

Answer 5

By deriving u in terms of x and setting the focus to the du.

Question 6

What must happen in relation to the inner function when Integrating a e^x function.

Answer 6

e^x will be divided by the derivative of the inner function, that being the power.

Question 7

What method should be used when Integrating two functions that have been multiplied and have no other relationship?

Answer 7

Integration by parts.

Question 8

In terms of ax^n what is the anti-derivative?

Answer 8

a/n+1 * x^n+1

Question 9

What method does one use when integrating a trig function squared?

Answer 9

Make use of an identity to turn the trig function into a different integratable trig function.

Question 10

What method does one use when an integral contains a square root with quadratic contents and no derivative in the intergrand?

Answer 10

Substitute the x with a trig function k,
then work out from there using identities, do not forget to apply the substitution rules.

Question 11

What are the rules of U-substitution?
(3)

Answer 11

-Derive u in terms of x to replace the dx.
-Rewrite the boundaries by calculating what the new variable would be were x the boundary Value.
-Rewrite it in terms of x at the end.

Question 12

What is the e^x integrated?

Answer 12

e^x + C

Question 13

What is a^x * ln a Integrated?

Answer 13

a^x + C

Question 14

What is 1/x integrated?

Answer 14

ln |x| +C

Question 15

When using Integration by parts what is always chosen as f(x)?

Answer 15

ln x

Question 16

When using integration by parts what is always chosen as g’(x)?

Answer 16

e^x

Question 17

When the boundaries of a definite Integral swap what must occur?

Answer 17

The sign of the Integral must also swap

Question 18

What can be done if there is a distributed constant within an Integral?

Answer 18

The constant can be “factorised” out of the Integral

Question 19

How can the Area between two curves be calculated?

Answer 19

Through integrating the Upper Function minus the Lower Function.
The intercepts can be determined through normal calculation or are given.

Question 20

What method can be used when a fraction with a high degree in the denominator is in an Integral?

Answer 20

Partial Fractions can be used to split the fraction into simpler fractions.

Question 21

What method should be used when there is a square root with linear contents within the intergrand?

Answer 21

Substitute the entire square root with a variable, remember to follow the rules of u-substitution.

Question 22

What can be done if there are two clearly seperate functions within an Integral where one is the derivative of the inner function of the other?

Answer 22

The one that is the derivative of the inner function of the other can he “cancelled” out and the other one can be integrated normally.

Question 23

What are the steps to determining the volume of a graph were it rotated around the X-axis or Y-axis

Answer 23

-Multiply the integrand by PI and square the Function within before integrating
-Remember to have the boundaries as the correct cut off points.
-If it is being rotated around the Y-axis the function within the integral must be written with x as the subject.

Question 24

What is the Integral of sin x?

Answer 24

cos x

Question 25

In a Riemann Sum what is represented by delta xi?

Answer 25

The width of the rectangles.
delta xi = (b-a)/n

Question 26

In a Riemann Sum what is represented by xi?

Answer 26

The Function Value at every possible point, or the height of the rectangles.
xi = a + i delta xi

Question 27

In a Riemann Sum what is represented by xi?

Answer 27

The Function Value at every possible point, or the height of the rectangles.
xi = a + i delta xi

Question 28

What method would be used if given a e exponetial or similar integral where the derivative of the inner function can be seen?

Answer 28

U-Substitution, potentially with V-Substitution there after