Practice sheet

Fresh questions from the whole chapter, to do on paper.

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Evaluate $\displaystyle\int_1^{\infty} \dfrac{dx}{x^{ 4 } }$ .
Evaluate $\displaystyle\int_0^{\infty} e^{-4x}\,dx$ .
What is $\displaystyle\int_{ 9 }^{\infty} \dfrac{dx}{x}$ ?
1. It diverges: $\ln b \to \infty$
2. It converges, because $\dfrac{1}{x} \to 0$
3. It equals $\ln 9$
What is $\displaystyle\int_{ 6 }^{\infty} \dfrac{dx}{x}$ ?
1. It diverges: $\ln b \to \infty$
2. It converges, because $\dfrac{1}{x} \to 0$
3. It equals $\ln 6$
What is $\displaystyle\int_{ 1 }^{\infty} \dfrac{dx}{x}$ ?
1. It diverges: $\ln b \to \infty$
2. It converges, because $\dfrac{1}{x} \to 0$
3. It equals $\ln 1$
What is $\displaystyle\int_{ 7 }^{\infty} \dfrac{dx}{x}$ ?
1. It diverges: $\ln b \to \infty$
2. It converges, because $\dfrac{1}{x} \to 0$
3. It equals $\ln 7$
What is $\displaystyle\int_{ 2 }^{\infty} \dfrac{dx}{x}$ ?
1. It diverges: $\ln b \to \infty$
2. It converges, because $\dfrac{1}{x} \to 0$
3. It equals $\ln 2$
What is $\displaystyle\int_{ 3 }^{\infty} \dfrac{dx}{x}$ ?
1. It diverges: $\ln b \to \infty$
2. It converges, because $\dfrac{1}{x} \to 0$
3. It equals $\ln 3$
What is $\displaystyle\int_{ 4 }^{\infty} \dfrac{dx}{x}$ ?
1. It diverges: $\ln b \to \infty$
2. It converges, because $\dfrac{1}{x} \to 0$
3. It equals $\ln 4$
What is $\displaystyle\int_{ 5 }^{\infty} \dfrac{dx}{x}$ ?
1. It diverges: $\ln b \to \infty$
2. It converges, because $\dfrac{1}{x} \to 0$
3. It equals $\ln 5$
What is $\displaystyle\int_{ 8 }^{\infty} \dfrac{dx}{x}$ ?
1. It diverges: $\ln b \to \infty$
2. It converges, because $\dfrac{1}{x} \to 0$
3. It equals $\ln 8$
Evaluate $\displaystyle\int_0^{ 25 } \dfrac{dx}{\sqrt{x} }$ .
What is $\displaystyle\int_{-2}^{ 2 } \dfrac{dx}{x^2}$ ?
1. It diverges: $\dfrac{1}{x^2}$ is unbounded at $x = 0$
2. $\left[-\dfrac{1}{x}\right]_{-2}^{ 2 } = -\dfrac{2}{ 2 }$
3. 0, by symmetry
What is $\displaystyle\int_{-3}^{ 3 } \dfrac{dx}{x^2}$ ?
1. It diverges: $\dfrac{1}{x^2}$ is unbounded at $x = 0$
2. $\left[-\dfrac{1}{x}\right]_{-3}^{ 3 } = -\dfrac{2}{ 3 }$
3. 0, by symmetry
What is $\displaystyle\int_{-4}^{ 4 } \dfrac{dx}{x^2}$ ?
1. It diverges: $\dfrac{1}{x^2}$ is unbounded at $x = 0$
2. $\left[-\dfrac{1}{x}\right]_{-4}^{ 4 } = -\dfrac{2}{ 4 }$
3. 0, by symmetry
What is $\displaystyle\int_{-1}^{ 1 } \dfrac{dx}{x^2}$ ?
1. It diverges: $\dfrac{1}{x^2}$ is unbounded at $x = 0$
2. $\left[-\dfrac{1}{x}\right]_{-1}^{ 1 } = -\dfrac{2}{ 1 }$
3. 0, by symmetry
What is $\displaystyle\int_{-5}^{ 5 } \dfrac{dx}{x^2}$ ?
1. It diverges: $\dfrac{1}{x^2}$ is unbounded at $x = 0$
2. $\left[-\dfrac{1}{x}\right]_{-5}^{ 5 } = -\dfrac{2}{ 5 }$
3. 0, by symmetry
Evaluate $\displaystyle\int_0^1 8\ln x\,dx$ .
For which $p$ does $\displaystyle\int_{ 2 }^{\infty} \dfrac{dx}{x^p}$ converge?
1. $p > 1$
2. $p \ge 1$
3. $p < 1$
For which $p$ does $\displaystyle\int_{ 3 }^{\infty} \dfrac{dx}{x^p}$ converge?
1. $p > 1$
2. $p \ge 1$
3. $p < 1$
For which $p$ does $\displaystyle\int_{ 9 }^{\infty} \dfrac{dx}{x^p}$ converge?
1. $p > 1$
2. $p \ge 1$
3. $p < 1$
For which $p$ does $\displaystyle\int_{ 5 }^{\infty} \dfrac{dx}{x^p}$ converge?
1. $p > 1$
2. $p \ge 1$
3. $p < 1$
For which $p$ does $\displaystyle\int_{ 7 }^{\infty} \dfrac{dx}{x^p}$ converge?
1. $p > 1$
2. $p \ge 1$
3. $p < 1$
For which $p$ does $\displaystyle\int_{ 4 }^{\infty} \dfrac{dx}{x^p}$ converge?
1. $p > 1$
2. $p \ge 1$
3. $p < 1$
For which $p$ does $\displaystyle\int_{ 1 }^{\infty} \dfrac{dx}{x^p}$ converge?
1. $p > 1$
2. $p \ge 1$
3. $p < 1$
For which $p$ does $\displaystyle\int_{ 8 }^{\infty} \dfrac{dx}{x^p}$ converge?
1. $p > 1$
2. $p \ge 1$
3. $p < 1$
For which $p$ does $\displaystyle\int_{ 6 }^{\infty} \dfrac{dx}{x^p}$ converge?
1. $p > 1$
2. $p \ge 1$
3. $p < 1$
Evaluate $\displaystyle\int_{ 3 }^{\infty} \dfrac{ 5 }{x^3}\,dx$ .

Answers

$\frac{1}{3}$
$\frac{1}{4}$
It diverges: $\ln b \to \infty$
$10$
It diverges: $\dfrac{1}{x^2}$ is unbounded at $x = 0$
$-8$
$p > 1$
$\frac{5}{18}$