<p>This method is used in cases of non-standard integrals. It introduces a suitable substitution to simplify a ‘complicated' term in the integrand. Here are some suggestions of possible substitutions to integrate the non-standard integrals:</p>
<p>Let \(a \) be a positive constant.</p>
<div class="content-block content-block--table"><div class="table-container"><table><caption>Suggestions for integration by substitution</caption><thead><tr><th style="text-align: left; --cell-justify: flex-start">When a function contains</th><th style="text-align: left; --cell-justify: flex-start">Try substituting</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">\(\sqrt{\mathrm{f}(x)}\)</td><td style="text-align: left; --cell-justify: flex-start">\(u=\mathrm{f}(x)\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(\ln x\)</td><td style="text-align: left; --cell-justify: flex-start">\(u=\ln x\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(e^x\)</td><td style="text-align: left; --cell-justify: flex-start">\(u=e^x\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(\sqrt{a^2-x^2}\)</td><td style="text-align: left; --cell-justify: flex-start">\(x=a \sin \theta\) or \(x=a \cos \theta\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(a^2+x^2\) or \(\sqrt{a^2+x^2}\)</td><td style="text-align: left; --cell-justify: flex-start">\(x=a \tan \theta\)</td></tr><tr><td style="text-align: left; --cell-justify: flex-start">\(\sqrt{x^2-a^2}\)</td><td style="text-align: left; --cell-justify: flex-start">\(x=a \sec \theta\)</td></tr></tbody></table></div></div>
<p>It is important that a substitution is legitimate if it corresponds to a bijective (one-one and onto) function defined over the domain of the integrand.  Such an assumption is often implicitly made when a substitution is suggested.</p>
<p><strong>Remark</strong>.</p>
<p>The method of substitution was first introduced by the German mathematician Karl Weierstrass (31 October 1815 – 19 February 1897).</p>

This method is used in cases of non-standard integrals. It introduces a suitable substitution to simplify a ‘complicated' term in the integrand. Here are some suggestions of possible substitutions to integrate the non-standard integrals:

It is important that a substitution is legitimate if it corresponds to a bijective (one-one and onto) function defined over the domain of the integrand.  Such an assumption is often implicitly made when a substitution is suggested.

The method of substitution was first introduced by the German mathematician Karl Weierstrass (31 October 1815 – 19 February 1897).

Integration by substitution

Integration by substitution

When a function contains	Try substituting
\(\sqrt{\mathrm{f}(x)}\)	\(u=\mathrm{f}(x)\)
\(\ln x\)	\(u=\ln x\)
\(e^x\)	\(u=e^x\)
\(\sqrt{a^2-x^2}\)	\(x=a \sin \theta\) or \(x=a \cos \theta\)
\(a^2+x^2\) or \(\sqrt{a^2+x^2}\)	\(x=a \tan \theta\)
\(\sqrt{x^2-a^2}\)	\(x=a \sec \theta\)