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<ol class="abc-list"><li data-list-item-id="item-1759046228915-ktfg1mpts" data-list-item-path="item-1759046228915-ktfg1mpts"><div class="list-item-paragraph">If&nbsp;$1 &lt; a &lt; b$, solve $\displaystyle\displaystyle \displaystyle\int_{0}^{b}{\left| a-{{x}^{2}} \right|}\mathrm{ d}x$, leaving your answer in terms of $a$ and $b$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759046229830-bwl4ix8g3" data-list-item-path="item-1759046229830-bwl4ix8g3"><div class="list-item-paragraph">Without the use of a graphic calculator, find the largest integer value of $n$ such that&nbsp;$\displaystyle\displaystyle \displaystyle\int_{\frac{n}{\mathrm{e}}}^{{{\mathrm{e}}^{\mathrm{e}}}}{\frac{1}{x{{\left( \ln x \right)}^{2}}}}\mathrm{ d}x$, where&nbsp;$1 &lt; \mathrm{e} &lt; n$, is positive.<span class="inline-marks">[4]</span></div></li></ol>

If $1 < a < b$, solve $\displaystyle\displaystyle \displaystyle\int_{0}^{b}{\left| a-{{x}^{2}} \right|}\mathrm{ d}x$, leaving your answer in terms of $a$ and $b$.

Without the use of a graphic calculator, find the largest integer value of $n$ such that $\displaystyle\displaystyle \displaystyle\int_{\frac{n}{\mathrm{e}}}^{{{\mathrm{e}}^{\mathrm{e}}}}{\frac{1}{x{{\left( \ln x \right)}^{2}}}}\mathrm{ d}x$, where $1 < \mathrm{e} < n$, is positive.

Integrals of Exponential, Logarithmic and Inverse Trigonometric Functions

Integrals involving Modulus Functions

Reverse Chain Rule with Logarithmic and Radical Inner Functions

2022 ACJC J2 MYE Q2

Techniques of Integration

Applications of Integration

<p>Find the following integrals. $\displaystyle\int{\frac{1}{x({{x}^{4}}+1)}\,\mathrm{d}x}\,;\,u={{x}^{2}}$</p>

Find the following integrals. $\displaystyle\displaystyle \displaystyle\int{\frac{1}{x({{x}^{4}}+1)}\,\mathrm{d}x}\,;\,u={{x}^{2}}$

Integration Using Partial Fractions

Integration by Substitution

ACJC Assignment 21 Q1 (c)

<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Evaluate $\displaystyle\displaystyle \displaystyle\int{\,\frac{1}{\sqrt{x}{{\left( 1+\sqrt{x} \right)}^{2}}}\mathrm{d}x}$.</div></li><li><div class="list-item-paragraph">Evaluate $\displaystyle\displaystyle \displaystyle\int{\frac{4}{{{e}^{-x}}-1}\,\mathrm{d}x}$.</div></li><li><div class="list-item-paragraph">Evaluate $\displaystyle\displaystyle \displaystyle\int{{{\left( \sin x+2\cos x \right)}^{2}}}\,\mathrm{d}x$.</div></li><li><div class="list-item-paragraph">Evaluate $\displaystyle\displaystyle \displaystyle\int{\sec \left( 2x-5 \right)\mathrm{ d}x}$.</div></li></ol>

Evaluate $\displaystyle\displaystyle \displaystyle\int{\,\frac{1}{\sqrt{x}{{\left( 1+\sqrt{x} \right)}^{2}}}\mathrm{d}x}$.

Evaluate $\displaystyle\displaystyle \displaystyle\int{\frac{4}{{{e}^{-x}}-1}\,\mathrm{d}x}$.

Evaluate $\displaystyle\displaystyle \displaystyle\int{{{\left( \sin x+2\cos x \right)}^{2}}}\,\mathrm{d}x$.

Evaluate $\displaystyle\displaystyle \displaystyle\int{\sec \left( 2x-5 \right)\mathrm{ d}x}$.

Integrating Powers of Sine and Cosine

Integrable Forms of Trigonometric Functions

Indefinite Integrals by Inspection

Logarithmic Integrals of $\frac{f'(x)}{f(x)}$

ACJC Assignment 20 Q1

<p>$\displaystyle\int{\frac{1+{{e}^{-x}}}{1-4{{e}^{-x}}}\mathrm{ d}x}$</p>

$\displaystyle\displaystyle \displaystyle\int{\frac{1+{{e}^{-x}}}{1-4{{e}^{-x}}}\mathrm{ d}x}$

ACJC Techniques of Integration Tutorial Q16

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that $\sqrt{\frac{1-2x}{1+2x}}$ can be written in the form $\frac{\mathrm{f}\left( x \right)}{\sqrt{1-4{{x}^{2}}}}$ where $\mathrm{f}\left( x \right)$ is a polynomial to be determined. Hence, find $\displaystyle\int{\sqrt{\frac{1-2x}{1+2x}}\,\mathrm{d}x}$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Show that $\displaystyle\int_{\mathrm{e}}^{{{\mathrm{e}}^{2}}}{\frac{1}{x\ln x}}\mathrm{ d}x=\ln 2$.</span></div><p style="text-align: right;">[3]</p></li></ol>

Show that $\sqrt{\frac{1-2x}{1+2x}}$ can be written in the form $\frac{\mathrm{f}\left( x \right)}{\sqrt{1-4{{x}^{2}}}}$ where $\mathrm{f}\left( x \right)$ is a polynomial to be determined. Hence, find $\displaystyle\displaystyle \displaystyle\int{\sqrt{\frac{1-2x}{1+2x}}\,\mathrm{d}x}$.

Show that $\displaystyle\displaystyle \displaystyle\int_{\mathrm{e}}^{{{\mathrm{e}}^{2}}}{\frac{1}{x\ln x}}\mathrm{ d}x=\ln 2$.

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