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<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-paragraph">Differentiate ${{\sin }^{-1}}\left( {{\mathrm{e}}^{2x}} \right)$ with respect to $x$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Hence find $\displaystyle\int{\frac{2{{\mathrm{e}}^{2x}}+{{\mathrm{e}}^{4x}}}{\sqrt{1-{{\mathrm{e}}^{4x}}}}}\,\mathrm{d}x$.</div><p style="text-align: right;">[3]</p></li></ol></li><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-paragraph">Show that ${{\sin }^{n-2}}\theta {{\cos }^{2}}\theta ={{\sin }^{n-2}}\theta -{{\sin }^{n}}\theta $, where $n\in {{\mathbb{Z}}^{+}}$, $n\ge 2$.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">By considering ${{\sin }^{n}}\theta ={{\sin }^{n-1}}\theta \sin \theta $, show that</div><div class="list-item-paragraph" style="text-align: center">$\displaystyle\int_{0}^{\frac{\pi }{2}}{{{\sin }^{n}}\theta \,\,\mathrm{d}\theta =\frac{n-1}{n}\displaystyle\int_{0}^{\frac{\pi }{2}}{{{\sin }^{n-2}}\theta \,\,\mathrm{d}\theta }}$,</div><div class="list-item-paragraph">where $n\in {{\mathbb{Z}}^{+}}$, $n\ge 2$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Hence find the exact value of $\displaystyle\int_{0}^{\frac{\pi }{2}}{{{\sin }^{3}}\theta \,\,\mathrm{d}\theta }$.</div><p style="text-align: right;">[2]</p></li></ol></li></ol>

Reduction Formulae for Integrals

Integration by Parts

Derivatives of Inverse Trigonometric Functions

Integration by the Reverse Chain Rule

Integrable Forms of Trigonometric Functions

Integration as an Inverse Process of Differentiation

Derivatives of Exponential Functions

Product, Quotient and Chain Rules

Chain Rule

2023 MI P1 Q10

Techniques of Integration

Applications of Integration

Techniques of Differentiation

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find $\displaystyle\int{\frac{6\tan 3x}{1+\cos 6x}\mathrm{d}x}$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">It is given that ${{I}_{n}}=\displaystyle\int_{1}^{\mathrm{e}}{{{\left( \ln x \right)}^{n}}\mathrm{d}x}$, for $n\in {{\mathbb{Z}}^{+}}$.</div><ol class="i-list"><li><div class="list-item-paragraph">Use integration by parts to show that</div><div class="list-item-paragraph" style="text-align: center">${{I}_{n}}=\mathrm{e}-n{{I}_{n-1}}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">The region bounded by the curve $y={{\left( \ln x \right)}^{2}}$, the $x$-axis and the line $x=\mathrm{e}$ is rotated through $2\pi $ radians about the $x$-axis. Using the result in <strong>(b)(i)</strong>, find the exact volume of the solid formed.</div><p style="text-align: right;">[3]</p></li></ol></li></ol>

Find $\displaystyle\displaystyle \displaystyle\int{\frac{6\tan 3x}{1+\cos 6x}\mathrm{d}x}$.

It is given that ${{I}_{n}}=\displaystyle\displaystyle \displaystyle\int_{1}^{\mathrm{e}}{{{\left( \ln x \right)}^{n}}\mathrm{d}x}$, for $n\in {{\mathbb{Z}}^{+}}$.

Volume of Revolution about the x-axis

2021 HCI P1 Q4

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that ${{I}_{n}}=\displaystyle\int_{1}^{\mathrm{e}}{x{{\left( \ln x \right)}^{n}}\,\mathrm{d}x}$ for $n\in \mathbb{Z}$, $n\ge 0$, show that</span></div><div class="list-item-paragraph" style="text-align: center">${{I}_{n}}=\frac{{{\mathrm{e}}^{2}}}{2}-\frac{n}{2}{{I}_{n-1}}$</div><div class="list-item-paragraph">for all $n\in {{\mathbb{Z}}^{+}}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Find the exact volume of the solid generated when the region bounded by the curve $y=\sqrt{x}\ln x$, and the $x$-axis and the line $x=\mathrm{e}$ is rotated completely about the $x$-axis.</span></div><p style="text-align: right;">[3]</p></li></ol>

Given that ${{I}_{n}}=\displaystyle\displaystyle \displaystyle\int_{1}^{\mathrm{e}}{x{{\left( \ln x \right)}^{n}}\,\mathrm{d}x}$ for $n\in \mathbb{Z}$, $n\ge 0$, show that

Find the exact volume of the solid generated when the region bounded by the curve $y=\sqrt{x}\ln x$, and the $x$-axis and the line $x=\mathrm{e}$ is rotated completely about the $x$-axis.

2022 NJC P1 Q1

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