Integration flashcards

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What is a reduction formula?

A formula that expresses an integral involving a parameter 'n' (e.g., $I_n$) in terms of an integral of the same form but with a lower value of 'n' (e.g., $I_{n-1}$ or $I_{n-2}$).

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What is the formula for integration by parts?

$\int u \frac{dv}{dx} \, dx = uv - \int v \frac{du}{dx} \, dx$. For definite integrals: $\int_a^b u \frac{dv}{dx} \, dx = [uv]_a^b - \int_a^b v \frac{du}{dx} \, dx$.

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How do you typically choose 'u' and 'dv/dx' when deriving a reduction formula for $\int x^n e^x \, dx$?

Choose $u = x^n$ and $\frac{dv}{dx} = e^x$. Differentiating $x^n$ reduces the power of x, which is the goal of a reduction formula.

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What is the formula for the mean value of a function $f(x)$ on the interval $[a, b]$?

The mean value $\bar{f}$ is given by $\frac{1}{b-a} \int_a^b f(x) \, dx$.

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What is the formula for the arc length of a curve $y=f(x)$ from $x=a$ to $x=b$?

$s = \int_a^b \sqrt{1 + (\frac{dy}{dx})^2} \, dx$.

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What is the formula for the arc length of a curve defined parametrically by $x=x(t), y=y(t)$ from $t=t_1$ to $t=t_2$?

$s = \int_{t_1}^{t_2} \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \, dt$.

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What is the formula for the surface area generated by rotating the curve $y=f(x)$ about the x-axis?

$A = \int_a^b 2\pi y \sqrt{1 + (\frac{dy}{dx})^2} \, dx$. Remember $2\pi y$ is the circumference of the circular strip.

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Common Trap: When using a reduction formula for a definite integral, what is a frequent mistake?

Forgetting to evaluate the '$uv$' part of integration by parts at the limits of integration. This term is often zero, but you must check and show it.

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What are the base cases $I_0$ and $I_1$ for the integral $I_n = \int_0^{\pi/2} \cos^n x \, dx$?

$I_0 = \int_0^{\pi/2} 1 \, dx = \frac{\pi}{2}$. $I_1 = \int_0^{\pi/2} \cos x \, dx = [\sin x]_0^{\pi/2} = 1$.

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Trap: In an arc length problem, the expression under the square root looks horrible. What should I check for?

The expression inside the square root, e.g., $1 + (\frac{dy}{dx})^2$, often simplifies to a perfect square. Always try to factorise it before attempting to integrate.