Write down the integral to be evaluated: I=_1^1x^2 dx State why the integral is improper: improper: unbounded interval State the final answer: I=1 Answer: 1

Write down the integral to be evaluated: I=_2^1x^3 dx State why the integral is improper: improper: unbounded interval State the final answer: I=(1)/(8) Answer: (1)/(8)

Write down the integral to be evaluated: I=_1^1x^4 dx State why the integral is improper: improper: unbounded interval Replace the offending limit by t and take a limit: _1^1x^4 dx=_t_1^t1x^4 dx State the final answer: I=(1)/(3) Answer: (1)/(3)

Evaluate _0^91√(x) dx.

Write down the integral to be evaluated: I=_0^91√(x) dx State why the integral is improper: improper: f(x) as x0 State the final answer: I=6 Answer: 6

Evaluate _0^41√(x) dx.

Write down the integral to be evaluated: I=_0^41√(x) dx State why the integral is improper: improper: f(x) as x0 State the final answer: I=4 Answer: 4

Methods in calculus Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Methods in calculus questions. See exactly how to solve problems on improper-integrals, power-rule, exponential-integration, mean-value.

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Further Maths70 questionsStep-by-step solutions

Question 1

2 markseasy

Evaluate

\int_{1}^{\infty}\frac{1}{x^{2}}\,dx

Worked solution

Write down the integral to be evaluated
$I=\int_{1}^{\infty}\frac{1}{x^{2}}\,dx$
Identify the integrand and the limits of integration.
State why the integral is improper
$\text{improper: }\text{unbounded interval}$
This integral is improper because the upper limit is infinite.
State the final answer
$I=1$
This is the exact value required.

Answer

1

Question 2

2 markseasy

Evaluate

\int_{2}^{\infty}\frac{1}{x^{3}}\,dx

Worked solution

Write down the integral to be evaluated
$I=\int_{2}^{\infty}\frac{1}{x^{3}}\,dx$
Identify the integrand and the limits of integration.
State why the integral is improper
$\text{improper: }\text{unbounded interval}$
This integral is improper because the upper limit is infinite.
State the final answer
$I=\frac{1}{8}$
This is the exact value required.

Answer

\frac{1}{8}

Question 3

2 markseasy

Evaluate

\int_{1}^{\infty}\frac{1}{x^{4}}\,dx

Worked solution

Write down the integral to be evaluated
$I=\int_{1}^{\infty}\frac{1}{x^{4}}\,dx$
Identify the integrand and the limits of integration.
State why the integral is improper
$\text{improper: }\text{unbounded interval}$
This integral is improper because the upper limit is infinite.
Replace the offending limit by $t$ and take a limit
$\int_{1}^{\infty}\frac{1}{x^{4}}\,dx=\lim_{t\to\infty}\int_{1}^{t}\frac{1}{x^{4}}\,dx$
The improper integral is DEFINED as this limit; it is not a substitution.
State the final answer
$I=\frac{1}{3}$
This is the exact value required.

Answer

\frac{1}{3}

Question 4

2 markseasy

Evaluate

\int_{0}^{9}\frac{1}{\sqrt{x}}\,dx

Worked solution

Write down the integral to be evaluated
$I=\int_{0}^{9}\frac{1}{\sqrt{x}}\,dx$
Identify the integrand and the limits of integration.
State why the integral is improper
$\text{improper: }f(x)\to\infty\text{ as }x\to0$
This integral is improper because the integrand is undefined at $x=0$ .
State the final answer
$I=6$
This is the exact value required.

Answer

6

Question 5

2 markseasy

Evaluate

\int_{0}^{4}\frac{1}{\sqrt{x}}\,dx

Worked solution

Write down the integral to be evaluated
$I=\int_{0}^{4}\frac{1}{\sqrt{x}}\,dx$
Identify the integrand and the limits of integration.
State why the integral is improper
$\text{improper: }f(x)\to\infty\text{ as }x\to0$
This integral is improper because the integrand is undefined at $x=0$ .
State the final answer
$I=4$
This is the exact value required.

Answer

4

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