9709 · 3.5
Integration flashcards
Revision flashcards for Cambridge 9709 Integration (syllabus 3.5). Flip, recall, then mark a real past-paper question.
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What is an indefinite integral?
The family of functions whose derivative is the integrand. It is written as $\int f(x) dx = F(x) + C$, where $F'(x) = f(x)$ and $C$ is the constant of integration.
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What does $\int_a^b f(x) dx$ represent geometrically?
The signed area between the curve $y=f(x)$, the x-axis, and the vertical lines $x=a$ and $x=b$. Area below the x-axis contributes a negative value to the integral.
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What is the integral of $e^{ax+b}$?
$\int e^{ax+b} dx = \frac{1}{a}e^{ax+b} + C$. A common mistake is forgetting the $\frac{1}{a}$ factor.
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What is the integral of $\frac{1}{ax+b}$?
$\int \frac{1}{ax+b} dx = \frac{1}{a}\ln|ax+b| + C$. Remember the $\frac{1}{a}$ factor and the modulus signs for the logarithm.
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What is the integration by parts formula?
$\int u \frac{dv}{dx} dx = uv - \int v \frac{du}{dx} dx$. It is used to integrate the product of two functions.
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What is the LIATE rule for?
It's a mnemonic to help choose 'u' in integration by parts. The order of preference is: Logarithmic, Inverse Trig, Algebraic, Trigonometric, Exponential.
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What must you do with limits in a definite integral when using substitution?
You must change the limits to be in terms of the new variable, 'u'. If the original limits are $x=a$ and $x=b$, find the corresponding values for $u$ and use those for the new integral.
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How do you integrate $\tan x$?
Rewrite it as $\int \frac{\sin x}{\cos x} dx$. This is of the form $-\int \frac{f'(x)}{f(x)} dx$, where $f(x)=\cos x$. The result is $-\ln|\cos x| + C$, which is equivalent to $\ln|\sec x| + C$.
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How do you integrate $\cos^2 x$ or $\sin^2 x$?
Use the double angle identities. For $\cos^2 x$, use $\cos(2x) = 2\cos^2 x - 1$ to get $\cos^2 x = \frac{1}{2}(1+\cos(2x))$. For $\sin^2 x$, use $\cos(2x) = 1 - 2\sin^2 x$ to get $\sin^2 x = \frac{1}{2}(1-\cos(2x))$.
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What is the formula for the volume of revolution about the x-axis?
$V = \pi \int_a^b y^2 dx$. Remember to square the function $y=f(x)$ *before* you integrate.
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What is the integral of the form $\int \frac{f'(x)}{f(x)} dx$?
This is a standard pattern that integrates to $\ln|f(x)| + C$. Recognising this saves you from using substitution.