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PSLE Foundation Mathematics

<p>$\displaystyle\displaystyle \displaystyle\int{\frac{\ln x}{x\sqrt{1-{{\left( \ln x \right)}^{2}}}}}\,\mathrm{d}x$</p>

$\displaystyle\displaystyle \displaystyle\int{\frac{\ln x}{x\sqrt{1-{{\left( \ln x \right)}^{2}}}}}\,\mathrm{d}x$

Reverse Chain Rule with Logarithmic and Radical Inner Functions

Techniques of Integration

<p>$\displaystyle\displaystyle \displaystyle\int{\frac{1}{\sqrt{x}{{\left( 1-\sqrt{x} \right)}^{5}}}}\,\mathrm{d}x$</p>

$\displaystyle\displaystyle \displaystyle\int{\frac{1}{\sqrt{x}{{\left( 1-\sqrt{x} \right)}^{5}}}}\,\mathrm{d}x$

<ol class="i-list"><li><div class="list-item-paragraph">$\displaystyle\int{\frac{1}{x{{\left( 1-\ln x \right)}^{5}}}\mathrm{d}x}$</div></li><li><div class="list-item-paragraph">$\displaystyle\int{\frac{1}{\sqrt{x}{{\left( 1-\sqrt{x} \right)}^{5}}}\mathrm{d}x}$</div></li></ol>

<p>$\displaystyle\int{\frac{1}{\sqrt{x}\left( {{e}^{\sqrt{x}}} \right)}}\mathrm{ d}x$</p>

$\displaystyle\displaystyle \displaystyle\int{\frac{1}{\sqrt{x}\left( {{e}^{\sqrt{x}}} \right)}}\mathrm{ d}x$

<p>$\displaystyle\int{\frac{1}{x\ln x}}\mathrm{ d}x$</p>

$\displaystyle\displaystyle \displaystyle\int{\frac{1}{x\ln x}}\mathrm{ d}x$

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

<p>$\displaystyle\int{\frac{\ln x}{x}}\mathrm{ d}x$</p>

$\displaystyle\displaystyle \displaystyle\int{\frac{\ln x}{x}}\mathrm{ d}x$

<p>$\displaystyle\int{\frac{{{\left( 5-\sqrt{x} \right)}^{10}}}{\sqrt{x}}}\mathrm{ d}x$</p>

$\displaystyle\displaystyle \displaystyle\int{\frac{{{\left( 5-\sqrt{x} \right)}^{10}}}{\sqrt{x}}}\mathrm{ d}x$

<p>$\displaystyle\int{\frac{{{\left( 3-\ln x \right)}^{4}}}{x}}\mathrm{ d}x$</p>

$\displaystyle\displaystyle \displaystyle\int{\frac{{{\left( 3-\ln x \right)}^{4}}}{x}}\mathrm{ d}x$

<p>Evaluate the following integrals. </p>
<ol class="abc-list"><li><div class="list-item-paragraph">${\displaystyle\int \frac{(\ln x)^4}{x} \mathrm{ d}x}$</div></li><li><div class="list-item-paragraph">${\displaystyle\int \frac{1}{x(1 - \ln x)} \mathrm{ d}x}$</div></li><li><div class="list-item-paragraph">${\displaystyle\int \frac{1}{x\sqrt{1 - \ln x}} \mathrm{ d}x}$</div></li><li><div class="list-item-paragraph">${\displaystyle\int \frac{(3 - \sqrt{x})^5}{\sqrt{x}} \mathrm{ d}x}$</div></li><li><div class="list-item-paragraph">${\displaystyle\int \frac{1}{\sqrt{x}(1 - \sqrt{x})} \mathrm{ d}x}$</div></li><li><div class="list-item-paragraph">${\displaystyle\int \frac{1}{\sqrt{x}(\sqrt{x} + 1)^5} \mathrm{ d}x}$</div></li></ol>

Method of substitution (Cancelling Method) Notes

<ol class="abc-list"><li data-list-item-id="item-1760430500657-1ac8nrbb5" data-list-item-path="item-1760430500657-1ac8nrbb5"><div class="list-item-paragraph">Show that&nbsp;$\frac{4{{x}^{2}}-6x+1}{\left( {{x}^{2}}+1 \right)\left( x-2 \right)}$ can be expressed as&nbsp;$\frac{Ax+B}{{{x}^{2}}+1}-\frac{C}{x-2}$, where&nbsp;$A$, $B$ and $C$ are constants to be determined. Hence find $\displaystyle \displaystyle\int{\frac{4{{x}^{2}}-6x+1}{\left( {{x}^{2}}+1 \right)\left( x-2 \right)}\mathrm{d}x}$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1760430501557-soilq958r" data-list-item-path="item-1760430501557-soilq958r"><div class="list-item-paragraph">Differentiate ${{\mathrm{e}}^{\sqrt{x}}}$ with respect to $x$. Hence, or otherwise, find $\displaystyle \displaystyle\int{\frac{{{\mathrm{e}}^{\sqrt{x}}}}{\sqrt{x}{{\left( {{e}^{\sqrt{x}}}+1 \right)}^{4}}}\mathrm{d}x}$.<span class="inline-marks">[3]</span></div></li></ol>

Integration Using Partial Fractions

Derivatives of Exponential Functions

Partial Fractions

Product, Quotient and Chain Rules

Chain Rule

Standard Integrals with Quadratic Denominators

2022 NYJC Promo Q2

Techniques of Differentiation

<p>A curve $C$ is defined by $y=\frac{{{\left( \ln x \right)}^{4}}}{\sqrt{x}}$ where $0 < x < 10$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the exact volume generated when the area bounded by $C$, the $x$-axis and the lines $x=1$ and $x=\mathrm{e}$ is rotated about the $x$-axis through ${{360}^{\circ }}$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Find the area enclosed by $C$ and the lines $y=1$ and $y=\mathrm{e}$.</div><p style="text-align: right;">[6]</p></li></ol>

A curve $C$ is defined by $y=\frac{{{\left( \ln x \right)}^{4}}}{\sqrt{x}}$ where $0 < x < 10$.

Volume of Revolution about the x-axis

Area bounded by 2 curves

2022 RI P1 Q8

Applications of Integration

<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

2022 ACJC J2 MYE Q2

<ol class="abc-list"><li data-list-item-id="item-1758965302705-n6b6rscpv" data-list-item-path="item-1758965302705-n6b6rscpv"><div class="list-item-paragraph">Find</div><ol class="i-list"><li data-list-item-id="subitem-1758965304790-jg3jgn8ne" data-list-item-path="item-1758965302705-n6b6rscpv/subitem-1758965304790-jg3jgn8ne"><div class="list-item-paragraph">$\displaystyle\displaystyle \displaystyle\int{\frac{1}{{{x}^{2}}+2x+5}\,\mathrm{d}x}$,<span class="inline-marks">[2]</span></div></li><li data-list-item-id="subitem-1758965305795-zofcuiqtc" data-list-item-path="item-1758965302705-n6b6rscpv/subitem-1758965305795-zofcuiqtc"><div class="list-item-paragraph">$\displaystyle\displaystyle \displaystyle\int{\frac{{{(\ln x)}^{3}}}{x}\,\mathrm{d}x}$.<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="item-1758965309207-g4n3ce639" data-list-item-path="item-1758965309207-g4n3ce639"><div class="list-item-paragraph">Find the exact value of $\displaystyle\displaystyle \displaystyle\int_{0}^{\frac{\pi }{2}}{x\cos x\,\mathrm{d}x}$.<span class="inline-marks">[3]</span></div></li></ol>

Find the exact value of $\displaystyle\displaystyle \displaystyle\int_{0}^{\frac{\pi }{2}}{x\cos x\,\mathrm{d}x}$.

Integration by Parts

Integration by Completing the Square

2020 VJC Promo Q2

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

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