Question:

<p>f(x) =&nbsp;<span class="math-tex">\(\frac {1}{x(x^4 + 1)}\)</span><br /> <span class="math-tex">\(\Rightarrow\)</span>&nbsp;<span class="math-tex">\(\int\limits_1^2 {f(x)} \)</span>dx =&nbsp;<span class="math-tex">\(\int\limits_1^2 {\frac{1}{{x({x^4} + 1)}}} \)</span>dx =&nbsp;<span class="math-tex">\(\int\limits_1^2 {\frac{{{x^3}}}{{{x^4}({x^4} + 1)}}} \)</span><br /> Let x⁴ + 1 = t&nbsp;<span class="math-tex">\(\Rightarrow\)</span>&nbsp;x³ dx =&nbsp;<span class="math-tex">\(\frac {1}{4}\)</span>dt<br /> <span class="math-tex">\(\Rightarrow\)</span>&nbsp;<span class="math-tex">\(\int\limits_1^2 {f(x)} \)</span>dx =&nbsp;<span class="math-tex">\(\frac{1}{4}\int\limits_2^{17} {\frac{1}{{(t - 1)t}}}\)</span>dt<br /> =&nbsp;<span class="math-tex">\(\frac{1}{4}\int\limits_2^{17} {\left( {\frac{1}{{t - 1)}} - \frac{1}{t}} \right)} \)</span>dt<br /> =&nbsp;<span class="math-tex">\(\frac{1}{4}\left[ {\log (t - 1) - \log t} \right]_2^{17}\)</span>&nbsp;-&nbsp;<span class="math-tex">\(\frac{1}{4}\left( {\log 1 - \log 2} \right)\)</span><br /> =&nbsp;<span class="math-tex">\(\frac{5}{4}\)</span>log 2 -&nbsp;<span class="math-tex">\(\frac{1}{4}\)</span>log 17<br /> =&nbsp;<span class="math-tex">\(\frac{1}{4}\)</span>log&nbsp;<span class="math-tex">\(\frac{32}{17}\)</span></p>