<ol class="abc-list"><li><ol class="i-list"><li><div class="list-item-paragraph">Express \(\frac{13x - 2}{(3 - x)(1 + 4x^2)}\) in partial fractions.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence, find the exact value of \(\int_{-\frac{1}{2}}^{\frac{1}{2}} \frac{13x - 2}{(3 - x)(1 + 4x^2)} \,\mathrm{d}x\), leaving your answer in the form \(\ln p + q\pi\), where&nbsp;\(p\) and \(q\) are rational numbers to be determined.<span class="inline-marks">[4]</span></div></li></ol></li><li><div class="list-item-paragraph">You are given that&nbsp;\(I = \int \frac{a^2 - x^2}{(a^2 + x^2)^2} \,\mathrm{d}x\), where&nbsp;\(a\) is a positive constant.</div><div class="list-item-paragraph">Use the substitution \(x = a \tan t\), where&nbsp;\(-\frac{\pi}{2} &lt; t &lt; \frac{\pi}{2}\), to find an expression for&nbsp;\(I\) in terms of \(a\), leaving your answer in a non-trigonometric form.<span class="inline-marks">[5]</span></div></li></ol>

You are given that \(I = \int \frac{a^2 - x^2}{(a^2 + x^2)^2} \,\mathrm{d}x\), where \(a\) is a positive constant.

Use the substitution \(x = a \tan t\), where \(-\frac{\pi}{2} < t < \frac{\pi}{2}\), to find an expression for \(I\) in terms of \(a\), leaving your answer in a non-trigonometric form.

Techniques of  Integration

International Baccalaureate Higher Level Mathematics

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