9709 · 2.5
Integration flashcards
Revision flashcards for Cambridge 9709 Integration (syllabus 2.5). Flip, recall, then mark a real past-paper question.
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What is an indefinite integral?
An integral without limits of integration, representing a family of functions. It always includes a constant of integration, `+c`.
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Why is the constant of integration, `+c`, necessary for indefinite integrals?
Because the derivative of a constant is zero. When reversing differentiation, we don't know what the original constant was, so we represent it with `+c`.
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What is the power rule for integration?
For $n \neq -1$, the integral of $x^n$ is $\int x^n \, dx = \frac{x^{n+1}}{n+1} + c$. You increase the power by one and divide by the new power.
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What is the integral of $\frac{1}{x}$?
$\int \frac{1}{x} \, dx = \ln|x| + c$. The modulus sign is important as the logarithm is only defined for positive inputs.
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What is a definite integral?
An integral with upper and lower limits, e.g., $\int_a^b f(x) \, dx$. It evaluates to a single numerical value, representing the signed area under the curve from $x=a$ to $x=b$.
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What does 'signed area' mean in the context of definite integrals?
Area above the x-axis is counted as positive, while area below the x-axis is counted as negative. The definite integral gives the net sum of these areas.
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How do you find the total area bounded by a curve and the x-axis if part of the curve is below the axis?
You must split the integral at the x-intercepts. Calculate the definite integral for each section separately and add the absolute values of the results.
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What is the formula for the Trapezium Rule?
$\int_a^b y \, dx \approx \frac{h}{2}[y_0 + y_n + 2(y_1 + y_2 + ... + y_{n-1})]$, where $h$ is the strip width and $y_i$ are the ordinates (y-values).
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How do you integrate functions of the form $(ax+b)^n$?
$\int (ax+b)^n \, dx = \frac{(ax+b)^{n+1}}{a(n+1)} + c$. Remember to divide by the coefficient of $x$, which is $a$.
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What are the integrals of $\sin(kx)$ and $\cos(kx)$?
$\int \sin(kx) \, dx = -\frac{1}{k}\cos(kx) + c$ and $\int \cos(kx) \, dx = \frac{1}{k}\sin(kx) + c$. Be careful with the sign change for the integral of sine.
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How do you find the area between two curves, $y=f(x)$ and $y=g(x)$?
First, find the points of intersection to determine the limits $a$ and $b$. Then, calculate $\int_a^b (y_{upper} - y_{lower}) \, dx$.
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When is the Trapezium Rule an overestimate?
When the curve is concave up (like a U-shape). The straight tops of the trapezia lie above the curve.