Improper Integrals (AQA A-Level Further Maths): Quizzes

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Improper Integrals
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Which characteristic makes an integral improper?

Infinite limits or undefined function

What defines a convergent improper integral?

Limit exists and equals finite number

How do you evaluate 2f(x)dx\int_2^{\infty} f(x) \, dx?

limt2tf(x)dx\lim_{t \to \infty} \int_2^{t} f(x) \, dx

For f(x)dx\int_{-\infty}^{\infty} f(x) \, dx split at 0, what must be true?

Both resulting integrals must converge

What is limxxkex\lim_{x \to \infty} x^k e^{-x} for any real kk?

00

What is limx0+xklnx\lim_{x \to 0^+} x^k \ln x for any real kk?

00

Why can't you substitute \infty directly into an antiderivative?

Not a valid mathematical operation

If f(x)f(x) is undefined at lower limit aa, what do you do?

Replace aa with tt, take limta\lim_{t \to a}

If f(x)f(x) is undefined at cc where a<c<ba < c < b, what must you do?

Split integral at cc

To find limxx2+12x2x\lim_{x \to \infty} \frac{x^2+1}{2x^2-x}, what technique works?

Divide numerator and denominator by x2x^2

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