Weekly challenge 45: Integral arranging

6094 /www/nrich/html/content/id/6094/ 1 2011-06-09T11:53:46 <mdoxml version="1.0"> <ul id="buttonBar"> <li> <a href="http://nrich.maths.org/6552&part=">Warm-up problem</a> </li> <li> <a href="http://nrich.maths.org/6502&part=">Try this next</a> </li> <li> <a href="https://nrich.maths.org/discus/messages/27/27.html?1307565044">Ask NRICH</a> </li> <li> <a href="http://integrals.wolfram.com/index.jsp">Read all about it</a> </li> <li> <a href="http://nrich.maths.org/6431/solution">Last week's solution</a> </li> </ul> <div style=""> How might you sort these integrals into an order or different groups? $$ \int\frac{1}{1+x^2}\rm dx\quad\quad\int\frac{1}{1-x^2}\,dx $$ $$ \int\frac{1}{(1+x)^2}\,dx\quad\quad\int\frac{1}{(1-x)^2}\,dx $$ $$ \int\frac{1}{1+x}\,dx\quad\quad\int\frac{1}{1-x}\,dx $$ $$ \int\frac{1}{\sqrt{1+x^2}}\,dx\quad\quad\int\frac{1}{\sqrt{1-x}}\,dx $$ $$ \int{\sqrt{1+x^2}}\,dx\quad\quad\int{\sqrt{1-x^2}}\,dx $$ $$ \int \sqrt{1+x}dx\quad\quad\int \sqrt{1-x}\,dx $$ </div> <div class="framework"> Did you know ... ? Although you can compute many integrals using <a href="http://integrals.wolfram.com/index.jsp">Wolfram's integrator</a>, if you do enough mathematics you will realise that the class of functions which integrate to a closed algebraic form is, by most ways of counting, small. There are many advanced analytical tools which allow for the manipulation and approximate computation of integrals more generally. A large part of this procedure involves classifying integrals into different types before suitable approximations are made. </div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">There is no particular 'answer' to this question, but here are some points to consider: <ul> <li>Are there some that you know the answer to, others that you don't?</li> <li>Are some equivalent under a change of variables?</li> <li>Are some valid over different ranges of integration?</li> <li>Are some larger than others over different ranges?</li> <li>Are some more complex than others to evaluate?</li> <li>Do some seem to have the same 'class' of answer?</li> </ul> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">There is no particular 'answer' to this question, but here are some points to consider: <ul> <li>Are there some that you know the answer to, others that you don't?</li> <li>Are some equivalent under a change of variables?</li> <li>Are some valid over different ranges of integration?</li> <li>Are some larger than others over different ranges?</li> <li>Are some more complex than others to evaluate?</li> <li>Do some seem to have the same 'class' of answer?</li> </ul> </mdoxml> 2 4 0 0 0 0 1 Weekly challenge 45: Integral arranging How would you sort out these integrals? Pre-Calculus and Calculus Calculus generally Pre-Calculus and Calculus Integration