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PSLE Foundation Mathematics

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Integrate $\frac{3(1 - x)^{3} - 2}{1 - x}$ with respect to $x$.</span></div></li></ol>

Integrate $\frac{3(1 - x)^{3} - 2}{1 - x}$ with respect to $x$.

Integration of Power Functions

Integration of Reciprocal Linear Functions

SCGS 4AM Prelim 2022

Integration

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find $\displaystyle\displaystyle\int \sqrt{5x + 3} \mathrm{d}x$ .</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Hence evaluate $\displaystyle\displaystyle\int_{0}^{2} \sqrt{5x + 3} \mathrm{d}x$ .</div></li></ol>

Find $\displaystyle\displaystyle\int \sqrt{5x + 3} \mathrm{d}x$ .

Hence evaluate $\displaystyle\displaystyle\int_{0}^{2} \sqrt{5x + 3} \mathrm{d}x$ .

Definite Integral

<p>Integrate the following expressions with respect to $x$.</p>
<ol class="abc-list"><li data-list-item-id="item-1770456207213-kvbmf61cb" data-list-item-path="item-1770456207213-kvbmf61cb"><div class="list-item-paragraph">${\displaystyle\displaystyle\int 2\pi \sin \left( \frac{2ex+3}{4} \right) \mathrm{d}x}$</div></li><li data-list-item-id="item-1770456207708-wm5uwy771" data-list-item-path="item-1770456207708-wm5uwy771"><div class="list-item-paragraph">${\displaystyle\displaystyle\int\frac{5-37x}{37x} \ \mathrm{d}x}$</div></li><li data-list-item-id="item-1770456208233-8j99a55ge" data-list-item-path="item-1770456208233-8j99a55ge"><div class="list-item-paragraph">${\displaystyle\int(3+x)\frac{7(3+x)+4}{(3+x)} \ \mathrm{d}x}$</div></li></ol>

Integrate the following expressions with respect to $x$.

Integrable Forms of Trigonometric Functions

<p>Integrate the following expressions with respect to $x$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">${\displaystyle\displaystyle\int 3\cos 7x + \sec^2 x \ \mathrm{d}x}$</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">${\displaystyle\displaystyle\int \frac{5}{x^4} - \cos(7x - 4) \ \mathrm{d}x}$</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">${\displaystyle\displaystyle\int \sin 5x - \sec^2 2x \ \mathrm{d}x}$</div></li></ol>

${\displaystyle\displaystyle\int 3\cos 7x + \sec^2 x \ \mathrm{d}x}$

${\displaystyle\displaystyle\int \frac{5}{x^4} - \cos(7x - 4) \ \mathrm{d}x}$

${\displaystyle\displaystyle\int \sin 5x - \sec^2 2x \ \mathrm{d}x}$

<p>${\displaystyle\displaystyle\int \frac{5\sqrt[3]{x} + 7x^3 + \sqrt{x}}{2x^4} \ \mathrm{d}x}$</p>

${\displaystyle\displaystyle\int \frac{5\sqrt[3]{x} + 7x^3 + \sqrt{x}}{2x^4} \ \mathrm{d}x}$

<p>${\displaystyle\displaystyle\int \frac{2x^4 - 8x^3 + \sqrt{x}}{(\sqrt{x})^3} \ \mathrm{d}x}$</p>

${\displaystyle\displaystyle\int \frac{2x^4 - 8x^3 + \sqrt{x}}{(\sqrt{x})^3} \ \mathrm{d}x}$

<p>Integrate the following expressions with respect to $x$.</p>
<ol class="abc-list"><li data-list-item-id="item-1770453687881-orj56nz0s" data-list-item-path="item-1770453687881-orj56nz0s"><div class="list-item-paragraph">$\displaystyle\displaystyle\int \frac{x^2+2x+5}{(x+1)^3}\,\mathrm{d}x$</div></li><li data-list-item-id="item-1770453689007-nnf3cjqgd" data-list-item-path="item-1770453689007-nnf3cjqgd"><div class="list-item-paragraph">$\displaystyle\displaystyle\int \frac{\sqrt{5-x}+7}{\sqrt{5-x}}\,\mathrm{d}x$</div></li></ol>

<p>Integrate the following expressions with respect to $x$.</p>
<ol class="abc-list"><li data-list-item-id="item-1770452324724-e9jezct36" data-list-item-path="item-1770452324724-e9jezct36"><div class="list-item-paragraph">${\displaystyle\displaystyle\int \sqrt[3]{(9+3x)^5} \ \mathrm{d}x}$</div></li><li data-list-item-id="item-1770452325451-nn3pquy0x" data-list-item-path="item-1770452325451-nn3pquy0x"><div class="list-item-paragraph">${\displaystyle\displaystyle\int \frac{7}{3(x+10)^4} \ \mathrm{d}x}$</div></li></ol>

<p>Integrate the following expressions with respect to $x$.</p>
<ol class="abc-list"><li data-list-item-id="item-1770451987816-f9kiu09re" data-list-item-path="item-1770451987816-f9kiu09re"><div class="list-item-paragraph">${\displaystyle\displaystyle\int \left( \frac{5}{(6x+3)^4} \right)^2 \ \mathrm{d}x}$</div></li><li data-list-item-id="item-1770451988649-vd2tlxzn4" data-list-item-path="item-1770451988649-vd2tlxzn4"><div class="list-item-paragraph">${\displaystyle\displaystyle\int (2x + \sqrt{x^5})(2x - \sqrt{x^5}) \ \mathrm{d}x}$</div></li></ol>

<p>Integrate the following expressions with respect to $x$.</p>
<ol class="abc-list"><li data-list-item-id="item-1770450601814-0eqamd3r7" data-list-item-path="item-1770450601814-0eqamd3r7"><div class="list-item-paragraph">${\displaystyle\displaystyle\int 5x^6 + 8 \ \mathrm{d}x }$</div></li><li data-list-item-id="item-1770450602511-wmna9yyhv" data-list-item-path="item-1770450602511-wmna9yyhv"><div class="list-item-paragraph">${\displaystyle\displaystyle\int 3\sqrt{x} - \frac{2}{\sqrt{x}} + 3 \ \mathrm{d}x}$</div></li></ol>

<p>Evaluate $\displaystyle\displaystyle \displaystyle\int{\,{{(3x+6)}^{-4}}\ \mathrm{d}x}$ with respect to $x$ expressing your answer using positive indices.</p>

Evaluate $\displaystyle\displaystyle \displaystyle\int{\,{{(3x+6)}^{-4}}\ \mathrm{d}x}$ with respect to $x$ expressing your answer using positive indices.

<p>Integrate the following expressions with respect to $x$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$\displaystyle\int \sin^2 4x + \sqrt{5x} \, \mathrm{d}x$</div></li><li><div class="list-item-paragraph">$\displaystyle\int \cos^2 (3x + 1) - \cos\ (3x+1)\,\mathrm{d}x$</div></li><li><div class="list-item-paragraph">$\displaystyle\int \tan^2 (1 - 2x) + \sqrt{1-2x}\,\mathrm{d}x$</div></li></ol>

Integrate the following expressions with respect to $x$.

Integrating Powers of Sine and Cosine

<p>Integrate the following expressions with respect to $x$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$\frac{\tan^2 2x(4\sin^2 2x-1)}{\sin^2 2x}$</div></li><li><div class="list-item-paragraph">$\sin\left(\frac{\pi+x}{4}\right)-4x^3+5$</div></li><li><div class="list-item-paragraph">$\pi \cos \left( \frac{\pi x-1}{3} \right)$</div></li></ol>

<p>Integrate the following expressions with respect to $x$:</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$7\cos 3x + \sec^2 x$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\frac{1}{x^3} - \sin(5x + 2)$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\sin 3x - \sec^2 6x$</span></div></li></ol>

Integrate the following expressions with respect to $x$:

<p>Integrate the following expressions with respect to $x$:</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$\displaystyle\displaystyle \displaystyle\int \frac{5}{6x} \, \mathrm{d}x$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\displaystyle\displaystyle \displaystyle\int \frac{1}{4x-3} \, \mathrm{d}x$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\displaystyle\displaystyle \displaystyle\int \frac{8+(4x-3)^5}{(4x-3)^6} \, \mathrm{d}x$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\displaystyle\displaystyle \displaystyle\int \frac{5}{2-5x} \, \mathrm{d}x$</span></div></li></ol>

$\displaystyle\displaystyle \displaystyle\int \frac{5}{6x} \, \mathrm{d}x$

$\displaystyle\displaystyle \displaystyle\int \frac{1}{4x-3} \, \mathrm{d}x$

$\displaystyle\displaystyle \displaystyle\int \frac{8+(4x-3)^5}{(4x-3)^6} \, \mathrm{d}x$

$\displaystyle\displaystyle \displaystyle\int \frac{5}{2-5x} \, \mathrm{d}x$

<p>Integrate the following expressions with respect to $x$:</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">$\displaystyle\displaystyle \displaystyle\int \frac{1}{3x} \, \mathrm{d}x$</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">$\displaystyle\displaystyle \displaystyle\int \frac{5}{3x-1} \, \mathrm{d}x$</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-content"><span class="list-item-text">$\displaystyle\displaystyle \displaystyle\int \frac{3-(2x+1)^2}{(2x+1)^3} \, \mathrm{d}x$</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">$\displaystyle\displaystyle \displaystyle\int \frac{\sqrt{3-x}+5}{\sqrt{3-x}^{3}} \, \mathrm{d}x$</div></li></ol>

$\displaystyle\displaystyle \displaystyle\int \frac{1}{3x} \, \mathrm{d}x$

$\displaystyle\displaystyle \displaystyle\int \frac{5}{3x-1} \, \mathrm{d}x$

$\displaystyle\displaystyle \displaystyle\int \frac{3-(2x+1)^2}{(2x+1)^3} \, \mathrm{d}x$

<p>Integrate the following expressions with respect to $x$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$\sqrt{(5-4x)^3}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\frac{2}{(4x-1)^6}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\frac{5}{2(3x-4)^5}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$(x+\sqrt{x^3})(x-\sqrt{x^3})$</span></div></li></ol>

<p>Integrate the following expressions with respect to $x$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">$10x^4 - 5x^2 + x + 3$</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">$6\sqrt[3]{x} - \frac{2}{\sqrt{x}} + 8x^3$</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-content"><span class="list-item-text">$\frac{4\sqrt[5]{x} + 3x^4 + \sqrt{x}}{6x^3}$</span></div></li></ol>

$6\sqrt[3]{x} - \frac{2}{\sqrt{x}} + 8x^3$

$\frac{4\sqrt[5]{x} + 3x^4 + \sqrt{x}}{6x^3}$

<p>Integrate the following expressions with respect to $x$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$3x^4+1$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$4\sqrt{x}-\frac{3}{\sqrt{x}}+1$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\frac{3x^3-4x^2+\sqrt{x}}{\sqrt{x}}$</span></div></li><li><div class="list-item-content"><span class="list-item-text">$\left(\frac{2}{(16x-1)^2}\right)^3$</span></div></li></ol>

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