Maths - Integration
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Flashcards
What is $x^n$ integrated?
What is $kx^n$ integrated?
Integrate $2x+4$?
What does $c$ mean in integration?
The constant of integration.
Why is a constant of integration important?
Because it represents any constant term that would disappear when being differentiated.
What does it mean to find a particular solution of an integral?
Calculating the actual value of an integral but solving for the constant of integration.
If a general solution specifies a family of curves, what does a particular solution specify?
A single curve.
If you have a general solution to an integral and is told that the solution passes through a point, what are you then finding?
A particular solution.
If $\int 2x+4 dx = x^2 + 4x + c$ but the actual solution passes through $(1,9)$, how can you solve for $c$?
What’s the flow chart process for finding a particular solution to an integral?
What is the first step for evaluating an integral $\int^b _ a$?
Finding the antiderivative/indefinite integral $F(x)$.
If $f(x)$ is a function, what is the notation for the antiderivative of $f(x)$?
When evaluating $\int^b _ a$, what do you do with $F(x)$?
How can you visualise $F(b) - F(a)$ when finding the area under a curve?
Finding the area up to the upper bound $b$ and then subtracting the unneccesary area up to $a$.
Why does the integral symbol look like $\int$?
It’s like a long S shape, representing a sum.
2021-06-09
\[\int \tan x dx\]
What is this equal to?
What’s another way of writing $\ln(\sec x) + c$?
\[\int \cot x dx\]
What is this equal to?
\[\int \sec x dx\]
What is this equal to?
\[\int \csc x dx\]
What is this equal to?
2021-07-03
What is $\sin^2(x)$ in terms of $\cos(2x)$?
What is $\cos^2(x)$ in terms of $\cos(2x)$?
How can you integrate $\cot^2 x$?
Rewrite as
\[\csc^2 x - 1\]2021-12-15
What is the formula for the area between the $x$-axis and a parametric curve defined with $x = f(t)$ and $y = g(t)$?
What is the formula for the area between the $y$-axis and a parametric curve defined with $x = f(t)$ and $y = g(t)$?
If you normally use $\int y \text{d}x$ when finding the area between the $x$-axis and a curve, how would this change for integrating parametrically?
If you normally use $\int x \text{d}y$ when finding the area between the $y$-axis and a curve, how would this change for integrating parametrically integrating parametrically?
2022-01-08
What is the integrand?
The inside of an integral, what is being integrated.
What’s the difference between the integral and the integrand?
The integrand is the function being integrated, whereas the integral is the whole expression.
2022-01-19
When doing a contextual integration question, what is dangerous?
Assuming that the object is around the origin.
What must you consider every time you do a contextual integration question where you’re modelling an object?
Where it is in relation to the origin.
2022-04-12
\[\int \sin(4x) (1 - \cos 4x)^3 \text{d}x\]
You could overcomplicate this by using several different trig identities and expanding. What could you also do?
Just notice that the derivative of the $\cos$ part is the four times the $\sin$ part.
2022-04-17
\[\int \frac{1-t^2}{1+t^2} \text{d} x\]
How should you tackle this?
Algebraic long division.
2022-05-11
\[\int \frac{3x}{\sqrt{4-x^2}} \text{d}x\]
How do you integrate this?
Consider
\[\frac{\text{d}}{\text{d}x} \sqrt{4-x^2}\]Why is
\[\int \frac{3x}{\sqrt{4-x^2}} \text{d}x\]
equal to
\[3\sqrt{4-x^2}\]
and not
\[-\frac{3}{2} \sqrt{4-x^2}\]
?
Because the $-\frac{1}{2}$ comes from the power rule and the chain rule.
2022-05-30
\[\int\frac{x^2}{1 + 16x^2}\text{d}x\]
How would you tackle this?
Algebraic long division.