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
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\[\int \tan x dx\]What is this equal to??
\[\ln(\sec x) + c\]\[\int \tan x dx\]
What is this equal to?
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\[\int \cot x dx\]What is this equal to??
\[\ln(\sin x) + c\]#####
\[\int \sec x dx\]What is this equal to??
\[\ln(\sec x + \tan x) + c\]#####
\[\int \csc x dx\]What is this equal to??
\[\ln(\csc x - \cot x) + c\]2021-07-03
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?
2021-12-15
\[\int \csc x dx\]
What is this equal to?
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
\[\csc^2 x - 1\]2022-01-08
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 $y = g(t)$?
2022-01-19
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?
2022-04-12
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\[\int \sin(4x) (1 - \cos 4x)^3 \text{d}x\]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
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\[\int \frac{1-t^2}{1+t^2} \text{d} x\]How should you tackle this?? Algebraic long division.
2022-05-11
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\[\int \frac{3x}{\sqrt{4-x^2}} \text{d}x\]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
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\[\int\frac{x^2}{1 + 16x^2}\text{d}x\]How would you tackle this?? Algebraic long division.
2022-06-06
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\[\lim _ {\delta x \to 0} \sum^6 _ {x = 2} \frac{1}{x} \delta x\]Can you write this as an integral??
\[\int^6_2 \frac{1}{x} \text{d}x\]#####
\[\int^10 _ 3 3x^2 - 4 \text{d}x\]Can you rewrite this as the limit of a sum??
\[\lim_{\delta x \to 0} \sum^{10}_{x \to 3} (3x^2 - 4) \delta x\]2022-06-07
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\[\sin 6x \sin 8x\]Rather than integrating by parts, how could you rewrite this in order to help with integration??
\[\frac{1}{2}\left( \cos(2x) - \cos(14x) \right)\]