Methods in Calculus (Edexcel A-Level Further Mathematics): Quizzes

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Improper Integrals
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When does an improper integral arise?

Integrand undefined or infinite range

Why is ∫021xdx\int_0^2 \frac{1}{\sqrt{x}} dx an improper integral?

1x\frac{1}{\sqrt{x}} undefined at x=0x = 0

Why is ∫1∞1x2dx\int_1^\infty \frac{1}{x^2} dx an improper integral?

Upper limit extends to infinity

How do you evaluate ∫021xdx\int_0^2 \frac{1}{\sqrt{x}} dx where integrand undefined at x=0x = 0?

Use lim⁡t→0+∫t2\lim_{t \to 0^+} \int_t^2

What is the antiderivative of 1x\frac{1}{\sqrt{x}}?

2x2\sqrt{x}

What is the value of ∫1∞1x2dx\int_1^\infty \frac{1}{x^2} dx?

11

What is the value of ∫0∞e−xdx\int_0^\infty e^{-x} dx?

11

For what values of pp does ∫1∞1xpdx\int_1^\infty \frac{1}{x^p} dx converge?

p>1p > 1

What is ∫0∞e−kxdx\int_0^\infty e^{-kx} dx for k>0k > 0?

1k\frac{1}{k}

Which is a common mistake with improper integrals?

Forgetting to evaluate the limit

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