A Math

A Math (IGCSE)

E Math

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

<ol class="i-list"><li data-list-item-id="amint-q02-question-1" data-list-item-path="amint-q02-question-1"><div class="list-item-paragraph">Express \(\dfrac{3x^2+3x+3}{(x-1)(x+2)^2}\) in partial fractions.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q02-question-2" data-list-item-path="amint-q02-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of \(\displaystyle\int_2^4 \dfrac{3x^2+3x+3}{(x-1)(x+2)^2}\,\mathrm{d}x\).<span class="inline-marks">[4]</span></div></li></ol>

Express \(\dfrac{3x^2+3x+3}{(x-1)(x+2)^2}\) in partial fractions.

Hence evaluate the exact value of \(\displaystyle\int_2^4 \dfrac{3x^2+3x+3}{(x-1)(x+2)^2}\,\mathrm{d}x\).

Problems involving Partial Fractions

Repeated-factor logarithmic integral

Integration

<ol class="i-list"><li data-list-item-id="amint-q18-question-1" data-list-item-path="amint-q18-question-1"><div class="list-item-paragraph">Show that&nbsp;\(\dfrac{\mathrm{d}}{\mathrm{d}x}\left[\mathrm{e}^{-2x}(\cos x+2\sin x)\right]=-5\mathrm{e}^{-2x}\sin x\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q18-question-2" data-list-item-path="amint-q18-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of \(\displaystyle\int_0^{\pi/4}\mathrm{e}^{-2x}\sin x\,\mathrm{d}x\).<span class="inline-marks">[4]</span></div></li></ol>

Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left[\mathrm{e}^{-2x}(\cos x+2\sin x)\right]=-5\mathrm{e}^{-2x}\sin x\).

Hence evaluate the exact value of \(\displaystyle\int_0^{\pi/4}\mathrm{e}^{-2x}\sin x\,\mathrm{d}x\).

Definite Integration as the Reverse of Differentiation

Exponential and trigonometric product

<ol class="i-list"><li data-list-item-id="amint-q07-question-1" data-list-item-path="amint-q07-question-1"><div class="list-item-paragraph">Show that&nbsp;\(\dfrac{\mathrm{d}}{\mathrm{d}x}\left[(x^2-2x+3)\sqrt{2x+1}\right]=\dfrac{5x^2-4x+1}{\sqrt{2x+1}}\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="amint-q07-question-2" data-list-item-path="amint-q07-question-2"><div class="list-item-paragraph">Hence evaluate \(\displaystyle\int_0^4\dfrac{5x^2-4x+1}{\sqrt{2x+1}}\,\mathrm{d}x\).<span class="inline-marks">[3]</span></div></li></ol>

Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left[(x^2-2x+3)\sqrt{2x+1}\right]=\dfrac{5x^2-4x+1}{\sqrt{2x+1}}\).

Hence evaluate \(\displaystyle\int_0^4\dfrac{5x^2-4x+1}{\sqrt{2x+1}}\,\mathrm{d}x\).

Quadratic factor with a square root

<ol class="i-list"><li data-list-item-id="amint-q03-question-1" data-list-item-path="amint-q03-question-1"><div class="list-item-paragraph">Express \(\dfrac{2x^3-3x^2-3x-6}{x(x-2)}\) in the form \(ax+b+\dfrac{c}{x}+\dfrac{d}{x-2}\), where&nbsp;\(a\), \(b\), \(c\) and \(d\) are integers.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q03-question-2" data-list-item-path="amint-q03-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of \(\displaystyle\int_3^5 \dfrac{2x^3-3x^2-3x-6}{x(x-2)}\,\mathrm{d}x\).<span class="inline-marks">[4]</span></div></li></ol>

Express \(\dfrac{2x^3-3x^2-3x-6}{x(x-2)}\) in the form \(ax+b+\dfrac{c}{x}+\dfrac{d}{x-2}\), where \(a\), \(b\), \(c\) and \(d\) are integers.

Hence evaluate the exact value of \(\displaystyle\int_3^5 \dfrac{2x^3-3x^2-3x-6}{x(x-2)}\,\mathrm{d}x\).

Improper rational definite integral

<ol class="i-list"><li data-list-item-id="amint-q13-question-1" data-list-item-path="amint-q13-question-1"><div class="list-item-paragraph">Differentiate \(x\tan x\) with respect to \(x\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amint-q13-question-2" data-list-item-path="amint-q13-question-2"><div class="list-item-paragraph">Differentiate \(\ln(\cos x)\) with respect to \(x\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amint-q13-question-3" data-list-item-path="amint-q13-question-3"><div class="list-item-paragraph">Hence evaluate the exact value of \(\displaystyle\int_0^{\pi/4}x\sec^2x\,\mathrm{d}x\).<span class="inline-marks">[3]</span></div></li></ol>

Differentiate \(x\tan x\) with respect to \(x\).

Differentiate \(\ln(\cos x)\) with respect to \(x\).

Hence evaluate the exact value of \(\displaystyle\int_0^{\pi/4}x\sec^2x\,\mathrm{d}x\).

Tangent product and logarithmic derivative

<ol class="i-list"><li data-list-item-id="amint-q21-question-1" data-list-item-path="amint-q21-question-1"><div class="list-item-paragraph">Show that&nbsp;\(\dfrac{\mathrm{d}}{\mathrm{d}x}[(2x+1)\ln x]=2\ln x+2+\dfrac1x\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q21-question-2" data-list-item-path="amint-q21-question-2"><div class="list-item-paragraph">Hence evaluate \(\displaystyle\int_1^{\mathrm{e}}\ln x\,\mathrm{d}x\).<span class="inline-marks">[4]</span></div></li></ol>

Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}[(2x+1)\ln x]=2\ln x+2+\dfrac1x\).

Hence evaluate \(\displaystyle\int_1^{\mathrm{e}}\ln x\,\mathrm{d}x\).

Logarithmic integral between one and e

<ol class="i-list"><li data-list-item-id="amint-q08-question-1" data-list-item-path="amint-q08-question-1"><div class="list-item-paragraph">Show that&nbsp;\(\dfrac{\mathrm{d}}{\mathrm{d}x}\left[(x^2+x+1)\sqrt[3]{x+1}\right]=\dfrac{7x^2+10x+4}{3(x+1)^{2/3}}\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="amint-q08-question-2" data-list-item-path="amint-q08-question-2"><div class="list-item-paragraph">Hence evaluate \(\displaystyle\int_0^7\dfrac{7x^2+10x+4}{3(x+1)^{2/3}}\,\mathrm{d}x\).<span class="inline-marks">[3]</span></div></li></ol>

Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left[(x^2+x+1)\sqrt[3]{x+1}\right]=\dfrac{7x^2+10x+4}{3(x+1)^{2/3}}\).

Hence evaluate \(\displaystyle\int_0^7\dfrac{7x^2+10x+4}{3(x+1)^{2/3}}\,\mathrm{d}x\).

Polynomial factor with a cube root

<ol class="i-list"><li data-list-item-id="amint-q09-question-1" data-list-item-path="amint-q09-question-1"><div class="list-item-paragraph">Show that&nbsp;\(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{3x+2}{\sqrt{x+4}}\right)=\dfrac{3x+22}{2(x+4)^{3/2}}\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="amint-q09-question-2" data-list-item-path="amint-q09-question-2"><div class="list-item-paragraph">Hence evaluate \(\displaystyle\int_{-3}^{5}\dfrac{3x+22}{2(x+4)^{3/2}}\,\mathrm{d}x\).<span class="inline-marks">[3]</span></div></li></ol>

Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{3x+2}{\sqrt{x+4}}\right)=\dfrac{3x+22}{2(x+4)^{3/2}}\).

Hence evaluate \(\displaystyle\int_{-3}^{5}\dfrac{3x+22}{2(x+4)^{3/2}}\,\mathrm{d}x\).

Linear numerator over a square root

<ol class="i-list"><li data-list-item-id="amint-q14-question-1" data-list-item-path="amint-q14-question-1"><div class="list-item-paragraph">Show that&nbsp;\(\dfrac{\mathrm{d}}{\mathrm{d}x}(\sin4x-4x)=-8\sin^22x\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q14-question-2" data-list-item-path="amint-q14-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of \(\displaystyle\int_0^{\pi/8}\sin^22x\,\mathrm{d}x\).<span class="inline-marks">[3]</span></div></li></ol>

Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}(\sin4x-4x)=-8\sin^22x\).

Hence evaluate the exact value of \(\displaystyle\int_0^{\pi/8}\sin^22x\,\mathrm{d}x\).

Squared-sine integral from a derivative identity

<ol class="i-list"><li data-list-item-id="amint-q10-question-1" data-list-item-path="amint-q10-question-1"><div class="list-item-paragraph">Differentiate \(x\cos3x\) with respect to \(x\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amint-q10-question-2" data-list-item-path="amint-q10-question-2"><div class="list-item-paragraph">Hence evaluate \(\displaystyle\int_0^{\pi/6}x\sin3x\,\mathrm{d}x\).<span class="inline-marks">[4]</span></div></li></ol>

Differentiate \(x\cos3x\) with respect to \(x\).

Hence evaluate \(\displaystyle\int_0^{\pi/6}x\sin3x\,\mathrm{d}x\).

Product derivative with a triple angle

<ol class="i-list"><li data-list-item-id="amint-q16-question-1" data-list-item-path="amint-q16-question-1"><div class="list-item-paragraph">Differentiate \((2x+1)\mathrm{e}^{-x}\) with respect to \(x\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="amint-q16-question-2" data-list-item-path="amint-q16-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of \(\displaystyle\int_0^1x\mathrm{e}^{-x}\,\mathrm{d}x\).<span class="inline-marks">[4]</span></div></li></ol>

Differentiate \((2x+1)\mathrm{e}^{-x}\) with respect to \(x\).

Hence evaluate the exact value of \(\displaystyle\int_0^1x\mathrm{e}^{-x}\,\mathrm{d}x\).

Linear factor with a negative exponential

<ol class="i-list"><li data-list-item-id="amint-q06-question-1" data-list-item-path="amint-q06-question-1"><div class="list-item-paragraph">Show that&nbsp;\(\dfrac{\mathrm{d}}{\mathrm{d}x}\left[(2x+5)\sqrt{3x-1}\right]=\dfrac{18x+11}{2\sqrt{3x-1}}\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q06-question-2" data-list-item-path="amint-q06-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of \(\displaystyle\int_1^3\dfrac{18x+11}{2\sqrt{3x-1}}\,\mathrm{d}x\).<span class="inline-marks">[3]</span></div></li></ol>

Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left[(2x+5)\sqrt{3x-1}\right]=\dfrac{18x+11}{2\sqrt{3x-1}}\).

Hence evaluate the exact value of \(\displaystyle\int_1^3\dfrac{18x+11}{2\sqrt{3x-1}}\,\mathrm{d}x\).

Scaled radical product

<ol class="i-list"><li data-list-item-id="amint-q05-question-1" data-list-item-path="amint-q05-question-1"><div class="list-item-paragraph">Show that&nbsp;\(\dfrac{\mathrm{d}}{\mathrm{d}x}\left[(3x-1)\sqrt{x+2}\right]=\dfrac{9x+11}{2\sqrt{x+2}}\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q05-question-2" data-list-item-path="amint-q05-question-2"><div class="list-item-paragraph">Hence evaluate \(\displaystyle\int_{-1}^{2}\dfrac{9x+11}{2\sqrt{x+2}}\,\mathrm{d}x\).<span class="inline-marks">[3]</span></div></li></ol>

Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left[(3x-1)\sqrt{x+2}\right]=\dfrac{9x+11}{2\sqrt{x+2}}\).

Hence evaluate \(\displaystyle\int_{-1}^{2}\dfrac{9x+11}{2\sqrt{x+2}}\,\mathrm{d}x\).

Linear factor with a square root

<ol class="i-list"><li data-list-item-id="amint-q17-question-1" data-list-item-path="amint-q17-question-1"><div class="list-item-paragraph">Differentiate \(\mathrm{e}^{2x}(3-2x)\) with respect to \(x\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q17-question-2" data-list-item-path="amint-q17-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of \(\displaystyle\int_0^1x\mathrm{e}^{2x}\,\mathrm{d}x\).<span class="inline-marks">[4]</span></div></li></ol>

Differentiate \(\mathrm{e}^{2x}(3-2x)\) with respect to \(x\).

Hence evaluate the exact value of \(\displaystyle\int_0^1x\mathrm{e}^{2x}\,\mathrm{d}x\).

Linear factor with a positive exponential

<ol class="i-list"><li data-list-item-id="amint-q19-question-1" data-list-item-path="amint-q19-question-1"><div class="list-item-paragraph">Show that&nbsp;\(\dfrac{\mathrm{d}}{\mathrm{d}x}\left[-(x^2+2x+2)\mathrm{e}^{-x}\right]=x^2\mathrm{e}^{-x}\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="amint-q19-question-2" data-list-item-path="amint-q19-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of \(\displaystyle\int_0^2x^2\mathrm{e}^{-x}\,\mathrm{d}x\).<span class="inline-marks">[3]</span></div></li></ol>

Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left[-(x^2+2x+2)\mathrm{e}^{-x}\right]=x^2\mathrm{e}^{-x}\).

Hence evaluate the exact value of \(\displaystyle\int_0^2x^2\mathrm{e}^{-x}\,\mathrm{d}x\).

Quadratic factor with a negative exponential

<ol class="i-list"><li data-list-item-id="amint-q12-question-1" data-list-item-path="amint-q12-question-1"><div class="list-item-paragraph">Show that&nbsp;\(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{\sin3x}{1-\cos3x}\right)=-\dfrac{3}{1-\cos3x}\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q12-question-2" data-list-item-path="amint-q12-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of \(\displaystyle\int_{\pi/12}^{\pi/6}\dfrac{1}{1-\cos3x}\,\mathrm{d}x\).<span class="inline-marks">[3]</span></div></li></ol>

Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{\sin3x}{1-\cos3x}\right)=-\dfrac{3}{1-\cos3x}\).

Hence evaluate the exact value of \(\displaystyle\int_{\pi/12}^{\pi/6}\dfrac{1}{1-\cos3x}\,\mathrm{d}x\).

Sine quotient with a triple angle

<ol class="i-list"><li data-list-item-id="amint-q15-question-1" data-list-item-path="amint-q15-question-1"><div class="list-item-paragraph">Show that&nbsp;\(\dfrac{\mathrm{d}}{\mathrm{d}x}(\sin3x-3x)=-6\sin^2\!\left(\dfrac{3x}{2}\right)\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q15-question-2" data-list-item-path="amint-q15-question-2"><div class="list-item-paragraph">Without using a calculator, find the values of the constants \(a\), \(b\) and \(c\) for which \(\displaystyle\int_0^{\pi/9}\left[\sin^2\!\left(\dfrac{3x}{2}\right)-\sin3x\right]\,\mathrm{d}x=a\sqrt3+b\pi+c\).<span class="inline-marks">[5]</span></div></li></ol>

Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}(\sin3x-3x)=-6\sin^2\!\left(\dfrac{3x}{2}\right)\).

Without using a calculator, find the values of the constants \(a\), \(b\) and \(c\) for which \(\displaystyle\int_0^{\pi/9}\left[\sin^2\!\left(\dfrac{3x}{2}\right)-\sin3x\right]\,\mathrm{d}x=a\sqrt3+b\pi+c\).

Problems involving Unknowns

Exact constants from a trigonometric integral

<ol class="i-list"><li data-list-item-id="amint-q01-question-1" data-list-item-path="amint-q01-question-1"><div class="list-item-paragraph">Express \(\dfrac{3x^2+3x+6}{x(x+1)(x+3)}\) in partial fractions.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q01-question-2" data-list-item-path="amint-q01-question-2"><div class="list-item-paragraph">Hence evaluate \(\displaystyle\int_1^2 \dfrac{3x^2+3x+6}{x(x+1)(x+3)}\,\mathrm{d}x\), giving your answer as a single logarithm.<span class="inline-marks">[4]</span></div></li></ol>

Express \(\dfrac{3x^2+3x+6}{x(x+1)(x+3)}\) in partial fractions.

Hence evaluate \(\displaystyle\int_1^2 \dfrac{3x^2+3x+6}{x(x+1)(x+3)}\,\mathrm{d}x\), giving your answer as a single logarithm.

Three-factor logarithmic integral

<ol class="i-list"><li data-list-item-id="amint-q04-question-1" data-list-item-path="amint-q04-question-1"><div class="list-item-paragraph">Express \(\dfrac{7x^2+6x+2}{(x+1)^2(2x-1)}\) in partial fractions.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q04-question-2" data-list-item-path="amint-q04-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of \(\displaystyle\int_1^2 \dfrac{7x^2+6x+2}{(x+1)^2(2x-1)}\,\mathrm{d}x\).<span class="inline-marks">[4]</span></div></li></ol>

Express \(\dfrac{7x^2+6x+2}{(x+1)^2(2x-1)}\) in partial fractions.

Hence evaluate the exact value of \(\displaystyle\int_1^2 \dfrac{7x^2+6x+2}{(x+1)^2(2x-1)}\,\mathrm{d}x\).

Repeated linear factors and an exact integral

<ol class="i-list"><li data-list-item-id="amint-q11-question-1" data-list-item-path="amint-q11-question-1"><div class="list-item-paragraph">Show that&nbsp;\(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{\sin2x}{1+\cos2x}\right)=\dfrac{2}{1+\cos2x}\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q11-question-2" data-list-item-path="amint-q11-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of \(\displaystyle\int_0^{\pi/6}\dfrac{1}{1+\cos2x}\,\mathrm{d}x\).<span class="inline-marks">[3]</span></div></li></ol>

Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{\sin2x}{1+\cos2x}\right)=\dfrac{2}{1+\cos2x}\).

Hence evaluate the exact value of \(\displaystyle\int_0^{\pi/6}\dfrac{1}{1+\cos2x}\,\mathrm{d}x\).

Math Question Bank