Integral ToolKit Flashcards
What is the most Important thing to never forget when answering an Integral?
+ C
In a Trig Integral what has to happen if there is an inner function?
The answer must be divided by the derivative of the inner function.
What alternative method should be used if a fraction can not be split into partial fractions?
U-Substitution
How would you integrate 3/x^2?
Using Normal Integration.
= -3/x
When using U-substitution how does one determine how the dx must change?
By deriving u in terms of x and setting the focus to the du.
What must happen in relation to the inner function when Integrating a e^x function.
e^x will be divided by the derivative of the inner function, that being the power.
What method should be used when Integrating two functions that have been multiplied and have no other relationship?
Integration by parts.
In terms of ax^n what is the anti-derivative?
a/n+1 * x^n+1
What method does one use when integrating a trig function squared?
Make use of an identity to turn the trig function into a different integratable trig function.
What method does one use when an integral contains a square root with quadratic contents and no derivative in the intergrand?
Substitute the x with a trig function k,
then work out from there using identities, do not forget to apply the substitution rules.
What are the rules of U-substitution?
(3)
-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.
What is the e^x integrated?
e^x + C
What is a^x * ln a Integrated?
a^x + C
What is 1/x integrated?
ln |x| +C
When using Integration by parts what is always chosen as f(x)?
ln x
When using integration by parts what is always chosen as g’(x)?
e^x
When the boundaries of a definite Integral swap what must occur?
The sign of the Integral must also swap
What can be done if there is a distributed constant within an Integral?
The constant can be “factorised” out of the Integral
How can the Area between two curves be calculated?
Through integrating the Upper Function minus the Lower Function.
The intercepts can be determined through normal calculation or are given.
What method can be used when a fraction with a high degree in the denominator is in an Integral?
Partial Fractions can be used to split the fraction into simpler fractions.
What method should be used when there is a square root with linear contents within the intergrand?
Substitute the entire square root with a variable, remember to follow the rules of u-substitution.
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?
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.
What are the steps to determining the volume of a graph were it rotated around the X-axis or Y-axis
-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.
What is the Integral of sin x?
cos x
In a Riemann Sum what is represented by delta xi?
The width of the rectangles.
delta xi = (b-a)/n
In a Riemann Sum what is represented by xi?
The Function Value at every possible point, or the height of the rectangles.
xi = a + i delta xi
In a Riemann Sum what is represented by xi?
The Function Value at every possible point, or the height of the rectangles.
xi = a + i delta xi
What method would be used if given a e exponetial or similar integral where the derivative of the inner function can be seen?
U-Substitution, potentially with V-Substitution there after
