Write down the integral to be evaluated: I=_3^1x^2 dx State why the integral is improper: the interval is unbounded above State the exact value: I=(1)/(3) Answer: (1)/(3)

Write down the integral to be evaluated: I=_1^2x^3 dx State why the integral is improper: the interval is unbounded above Replace the awkward limit by a variable and take a limit: _1^2x^3 dx=_t_1^t2x^3 dx State the exact value: I=1 Answer: 1

Write down the integral to be evaluated: I=_0^^-4x dx State why the integral is improper: the interval is unbounded above Replace the awkward limit by a variable and take a limit: _0^^-4x dx=_t_0^te^-4x dx State the exact value: I=(1)/(4) Answer: (1)/(4)

Evaluate _0^11√(x) dx.

Write down the integral to be evaluated: I=_0^11√(x) dx State why the integral is improper: f(x) as x0 State the exact value: I=2 Answer: 2

Evaluate _0^51x^2+25 dx. Give your answer in terms of π.

Write down the integral to be evaluated: I=_0^51x^2+25 dx Match the denominator with the standard form: a^2=25 a=5 Write down the antiderivative: F(x)=((x)/(5))5 State the exact value: I=(π)/(20) Answer: (π)/(20)

Further calculus Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Further calculus questions. See exactly how to solve problems on improper-integrals, power-rule, exponential-integration, inverse-tangent-integral.

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Further Maths70 questionsStep-by-step solutions

Question 1

2 markseasy

Evaluate

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

Worked solution

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

Answer

\frac{1}{3}

Question 2

2 markseasy

Evaluate

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

Worked solution

Write down the integral to be evaluated
$I=\int_{1}^{\infty}\frac{2}{x^{3}}\,dx$
Identify the integrand and the limits before choosing a method.
State why the integral is improper
$\text{the interval is unbounded above}$
This integral is improper because the upper limit of integration is infinite.
Replace the awkward limit by a variable and take a limit
$\int_{1}^{\infty}\frac{2}{x^{3}}\,dx=\lim_{t\to\infty}\int_{1}^{t}\frac{2}{x^{3}}\,dx$
The improper integral is DEFINED as this limit; replacing the offending limit by $t$ makes every step that follows a legitimate definite integral.
State the exact value
$I=1$
This is the exact answer required.

Answer

1

Question 3

2 markseasy

Evaluate

\int_{0}^{\infty}e^{-4x}\,dx

Worked solution

Write down the integral to be evaluated
$I=\int_{0}^{\infty}e^{-4x}\,dx$
Identify the integrand and the limits before choosing a method.
State why the integral is improper
$\text{the interval is unbounded above}$
This integral is improper because the upper limit of integration is infinite.
Replace the awkward limit by a variable and take a limit
$\int_{0}^{\infty}e^{-4x}\,dx=\lim_{t\to\infty}\int_{0}^{t}e^{-4x}\,dx$
The improper integral is DEFINED as this limit; replacing the offending limit by $t$ makes every step that follows a legitimate definite integral.
State the exact value
$I=\frac{1}{4}$
This is the exact answer required.

Answer

\frac{1}{4}

Question 4

2 markseasy

Evaluate

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

Worked solution

Write down the integral to be evaluated
$I=\int_{0}^{1}\frac{1}{\sqrt{x}}\,dx$
Identify the integrand and the limits before choosing a method.
State why the integral is improper
$f(x)\to\infty\ \text{as}\ x\to0$
This integral is improper because the integrand is unbounded at the lower limit $x=0$ .
State the exact value
$I=2$
This is the exact answer required.

Answer

2

Question 5

2 markseasy

Evaluate

\int_{0}^{5}\frac{1}{x^{2}+25}\,dx

. Give your answer in terms of

\pi

Worked solution

Write down the integral to be evaluated
$I=\int_{0}^{5}\frac{1}{x^{2}+25}\,dx$
Identify the integrand and the limits before choosing a method.
Match the denominator with the standard form
$a^{2}=25\quad\Rightarrow\quad a=5$
Reading off $a$ is all that is needed to quote the standard integral.
Write down the antiderivative
$F(x)=\frac{\arctan{\left(\frac{x}{5}\right)}}{5}$
This is the standard inverse-tangent result with $a=5$ .
State the exact value
$I=\frac{\pi}{20}$
This is the exact answer required.

Answer

\frac{\pi}{20}

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