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<p>Evaluate each of the following definite integrals.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">$\displaystyle\displaystyle\int_{3}^{6} 3x + \frac{2}{(x-5)^{2}} \mathrm{d}x$</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">$\displaystyle\displaystyle\int_{0}^{3} \left( \frac{x^{2} + 3x}{\sqrt{x}} \right) \mathrm{d}x$</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">$\displaystyle\displaystyle\int_{-1}^{3} x^{4} (x^{2} + 7x) \mathrm{d}x$</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">$\displaystyle\displaystyle\int_{0}^{1} \mathrm{e}^{2x} \mathrm{d}x$</div></li></ol>

Evaluate each of the following definite integrals.

$\displaystyle\displaystyle\int_{3}^{6} 3x + \frac{2}{(x-5)^{2}} \mathrm{d}x$

$\displaystyle\displaystyle\int_{0}^{3} \left( \frac{x^{2} + 3x}{\sqrt{x}} \right) \mathrm{d}x$

$\displaystyle\displaystyle\int_{-1}^{3} x^{4} (x^{2} + 7x) \mathrm{d}x$

$\displaystyle\displaystyle\int_{0}^{1} \mathrm{e}^{2x} \mathrm{d}x$

Improper Integrals and Convergence

Definite Integral

Integration

<p>Evaluate each definite integral.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">$\displaystyle\displaystyle\int_{-1}^{4} x^3 \ \mathrm{d}x$</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">$\displaystyle\displaystyle\int_{-\frac{1}{2}}^{\frac{1}{2}} (4 + 7x^2) \ \mathrm{d}x$</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">$\displaystyle\displaystyle\int_{2}^{5} \sqrt{y^3} \ \mathrm{d}y$</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">$\displaystyle\displaystyle\int_{-2}^{3} 2 + \frac{1}{\sqrt[3]{x}} \ \mathrm{d}x$</div></li></ol>

$\displaystyle\displaystyle\int_{-1}^{4} x^3 \ \mathrm{d}x$

$\displaystyle\displaystyle\int_{-\frac{1}{2}}^{\frac{1}{2}} (4 + 7x^2) \ \mathrm{d}x$

$\displaystyle\displaystyle\int_{2}^{5} \sqrt{y^3} \ \mathrm{d}y$

$\displaystyle\displaystyle\int_{-2}^{3} 2 + \frac{1}{\sqrt[3]{x}} \ \mathrm{d}x$

<ol class="abc-list"><li data-list-item-id="item-1770367385574-v6t4x6m44" data-list-item-path="item-1770367385574-v6t4x6m44"><div class="list-item-paragraph">Evaluate $\displaystyle\int_{0}^{\infty }{x{{e}^{-4x}}\,\mathrm{d}x}$.</div></li><li data-list-item-id="item-1770367386444-y7jy9glkx" data-list-item-path="item-1770367386444-y7jy9glkx"><div class="list-item-paragraph">Hence, find $\displaystyle\int_{0}^{\infty }{{{x}^{2}}{{e}^{-4x}}\,\mathrm{d}x}$.</div></li></ol>

Integration by Parts

Applications of Integration

Techniques of Integration

<p>Express $\displaystyle\int_{1}^{\infty }{x{{e}^{-2x}}\,\mathrm{d}x} $ in the form of $a+b{{e}^{-2}}$ where&nbsp;$a$ and $b$ are constants to be determined.</p>

Express $\displaystyle\int_{1}^{\infty }{x{{e}^{-2x}}\,\mathrm{d}x} $ in the form of $a+b{{e}^{-2}}$ where $a$ and $b$ are constants to be determined.

<p>By means of substitution $x = \tan \theta$, or otherwise, find the exact value of $\displaystyle\int_{0}^{\infty} \frac{x^{2}}{(1 + x^{2})^{2}}   dx$.</p>

By means of substitution $x = \tan \theta$, or otherwise, find the exact value of $\displaystyle\displaystyle \displaystyle\int_{0}^{\infty} \frac{x^{2}}{(1 + x^{2})^{2}}   dx$.

Integrating Powers of Sine and Cosine

Integrable Forms of Trigonometric Functions

Integration by Substitution

<p>Use the given substitution to find the exact numerical values of the definite integrals.</p>
<ol class="abc-list" start="5"><li><div class="list-item-paragraph">$\displaystyle\int_{0}^{\sqrt{3}}{\frac{{{x}^{2}}}{{{\left( 1+{{x}^{2}} \right)}^{2}}}\mathrm{d}x}$ $\qquad$ $x=\tan \theta $</div></li><li><div class="list-item-paragraph">$\displaystyle\int_{0}^{\frac{\sqrt{3}}{2}}{\sqrt{4-4{{x}^{2}}}\mathrm{d}x}$ $\qquad$ $x=\cos \theta $</div></li></ol>
<ol class="abc-list" start="9"><li><div class="list-item-paragraph">$\displaystyle\int_{0}^{\infty }{\frac{{{x}^{2}}}{1+{{x}^{6}}}\mathrm{d}x}$ $\qquad$ $\qquad$ $u={{x}^{3}}$</div></li><li><div class="list-item-paragraph">$\displaystyle\int_{0}^{a}{{{x}^{2}}\sqrt{{{a}^{2}}-{{x}^{2}}}\mathrm{d}x}$ $\qquad$ $x=a\sin \theta $, $a>0$</div></li></ol>

Use the given substitution to find the exact numerical values of the definite integrals.

Standard Integrals with Quadratic Denominators

NJC Integration Techniques Tutorial Q2

<ol class="abc-list"><li><div class="list-item-paragraph">The integral ${{I}_{n}}$ is defined by </div><div class="list-item-paragraph" style="text-align: center">\[
I_n = \displaystyle\int_{n\pi}^{(n+1)\pi} e^{-x}\,|\sin x|\,dx, for
\quad n \in \mathbb{N}.
\]</div><div class="list-item-paragraph">Show that</div><div class="list-item-paragraph" style="text-align: center">\[
I_0 = \frac{1}{2}\left( 1 + e^{-\pi} \right).
\]</div><p style="text-align: right;">[6]</p></li><li><div class="list-item-paragraph">By letting $y = x - n\pi$, show that</div><div class="list-item-paragraph" style="text-align: center">\[
I_n = e^{-n\pi} I_0.
\]</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Hence determine the exact value of $\displaystyle\int_{0}^{\infty }{{{\mathrm{e}}^{-x}}\left| \sin x \right|}\mathrm{ d}x$.</div><p style="text-align: right;">[5]</p></li></ol>

Integrals involving Modulus Functions

Cyclic Integration by Parts

Exponential Integral Involving Absolute Sine Function

<p>The integral test for convergence, known as the Maclaurin-Cauchy test, is a method used to test for convergence of an infinite series of non-negative terms. The Maclaurin-Cauchy test states that</p>
<p>‘For a non-negative and non-increasing function $\mathrm{f}$ defined on the interval $[1,\infty )$, a series of the form $\sum\limits_{r=1}^{\infty }{\mathrm{f}(r)}$ converges if $\displaystyle\int_{1}^{\infty }{\mathrm{f}(x)\,\mathrm{d}x}$ is finite.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that $\frac{{{\tan }^{-1}}x}{1+{{x}^{2}}}$ is positive and decreasing on the interval $[1,\infty )$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Use the Maclaurin-Cauchy test to determine if $\sum\limits_{r=1}^{\infty }{\frac{{{\tan }^{-1}}r}{1+{{r}^{2}}}}$ converges.</span></div><p style="text-align: right;">[5]</p></li></ol>

The integral test for convergence, known as the Maclaurin-Cauchy test, is a method used to test for convergence of an infinite series of non-negative terms. The Maclaurin-Cauchy test states that

‘For a non-negative and non-increasing function $\mathrm{f}$ defined on the interval $[1,\infty )$, a series of the form $\sum\limits_{r=1}^{\infty }{\mathrm{f}(r)}$ converges if $\displaystyle\displaystyle \displaystyle\int_{1}^{\infty }{\mathrm{f}(x)\,\mathrm{d}x}$ is finite.

Show that $\frac{{{\tan }^{-1}}x}{1+{{x}^{2}}}$ is positive and decreasing on the interval $[1,\infty )$.

Use the Maclaurin-Cauchy test to determine if $\sum\limits_{r=1}^{\infty }{\frac{{{\tan }^{-1}}r}{1+{{r}^{2}}}}$ converges.

Limit of a Sequence and Behaviour of a Sequence

Integration by the Reverse Chain Rule

2020 NJC P1 Q6

Functions

Sequence and Series

<ol class="i-list"><li><div class="list-item-paragraph">Using integration by parts, find $\displaystyle\int_{\mathrm{e}}^{n}{\frac{\ln x}{{{x}^{5}}}\mathrm{d}x}$ in terms of $n$ and $\mathrm{e}$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">The diagram below shows a part of the curve with equation $y=\frac{\ln x}{{{x}^{5}}}$. An <strong>infinite</strong> sequence of rectangles are formed using points on the curve as shown in the diagram.</div><div class="list-item-paragraph">Show that the total area of the rectangles is given by $\sum\limits_{k=3}^{\infty }{\frac{\ln k}{{{k}^{5}}}}$.<span class="inline-marks">[1]</span></div><div class="list-item-image"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/converted/1764584245965-4ypnqsrtrc7.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NjQ1ODQyNDU5NjUtNHlwbnFzcnRyYzcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjczMTU0OSwiZXhwIjoxNzkyNzMxODQ5fQ.NX5VF248L3Mk-zvEOsre5nftXXKhe4jOOmJBMrQ5PiY" alt="List item image" /></div></li><li><div class="list-item-paragraph">The Monotone Convergence Theorem for a series states that if ${{a}_{k}}>0$ for all $k$ and $\sum{{{a}_{k}}\le M}$ for some $M\in \mathbb{R}$, then $\sum{{{a}_{k}}}$ is convergent. By considering the integral $\displaystyle\int_{\mathrm{e}}^{\infty }{\frac{\ln x}{{{x}^{5}}}\mathrm{d}x}$ and the above graph, show that the series $\sum\limits_{k=3}^{\infty }{\frac{\ln k}{{{k}^{5}}}}$ is convergent.<span class="inline-marks">[4]</span></div></li></ol>

Using integration by parts, find $\displaystyle\displaystyle \displaystyle\int_{\mathrm{e}}^{n}{\frac{\ln x}{{{x}^{5}}}\mathrm{d}x}$ in terms of $n$ and $\mathrm{e}$.

The diagram below shows a part of the curve with equation $y=\frac{\ln x}{{{x}^{5}}}$. An **infinite** sequence of rectangles are formed using points on the curve as shown in the diagram.

The Monotone Convergence Theorem for a series states that if ${{a}_{k}}>0$ for all $k$ and $\sum{{{a}_{k}}\le M}$ for some $M\in \mathbb{R}$, then $\sum{{{a}_{k}}}$ is convergent. By considering the integral $\displaystyle\displaystyle \displaystyle\int_{\mathrm{e}}^{\infty }{\frac{\ln x}{{{x}^{5}}}\mathrm{d}x}$ and the above graph, show that the series $\sum\limits_{k=3}^{\infty }{\frac{\ln k}{{{k}^{5}}}}$ is convergent.

Riemann Sums

2019 NYJC CT2 P1 Q3

<p>Obtain a formula for&nbsp;$\displaystyle\displaystyle\int\limits_{\frac{1}{2}}^{n}{\frac{{{\left( {{\tan }^{-1}}2x \right)}^{2}}}{1+4{{x}^{2}}}}\mathrm{ d}x$ in terms of $n$, where&nbsp;$n&gt;0$.</p>
<p>Hence evaluate $\displaystyle\displaystyle\int\limits_{\frac{1}{2}}^{\infty }{\frac{{{\left( {{\tan }^{-1}}2x \right)}^{2}}}{1+4{{x}^{2}}}}\mathrm{ d}x$ exactly.<span class="inline-marks">[4]</span></p>

Obtain a formula for $\displaystyle\displaystyle\int\limits_{\frac{1}{2}}^{n}{\frac{{{\left( {{\tan }^{-1}}2x \right)}^{2}}}{1+4{{x}^{2}}}}\mathrm{ d}x$ in terms of $n$, where $n>0$.

Hence evaluate $\displaystyle\displaystyle\int\limits_{\frac{1}{2}}^{\infty }{\frac{{{\left( {{\tan }^{-1}}2x \right)}^{2}}}{1+4{{x}^{2}}}}\mathrm{ d}x$ exactly.

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