Integration By Substitution
Category : JEE Main & Advanced
(1) When integrand is a function i.e., \[\int{f\mathbf{[}\varphi \mathbf{(}x\mathbf{)}]\,\varphi '\mathbf{(}x\mathbf{)}\,dx}\]:
Here, we put \[\varphi (x)=t,\] so that \[\varphi '(x)dx=dt\] and in that case the integrand is reduced to \[\int{f(t)dt}\].
(2) When integrand is the product of two factors such that one is the derivative of the others i.e., \[I=\int{f(x).{f}'(x).dx}\]: In this case we put \[f(x)=t\] and convert it into a standard integral.
(3) Integral of a function of the form \[f\mathbf{(}ax+b\mathbf{)}\]: Here we put \[ax+b=t\] and convert it into standard integral. Obviously if \[\int{f(x)dx=\varphi (x),}\] then \[\int{f(ax+b)dx=\frac{1}{a}\varphi (ax+b)}+c\].
(4) If integral of a function of the form \[\frac{{f}'(x)}{f(x)}\]
\[\int_{{}}^{{}}{\frac{{f}'\text{(}x\text{)}}{f\text{(}x\text{)}}\,}dx=\log \,[f(x)]+c\]
(5) If integral of a function of the form \[{{[f(x)]}^{n}}{f}'(x)\] \[\int{{{[f\text{(}x\text{)}]}^{n}}{f}'\text{(}x\text{)}\,dx=\frac{{{[f\text{(}x\text{)}]}^{n+1}}}{n+1}}+c\], \[\text{ }\!\![\!\!\text{ }n\ne -1\text{ }\!\!]\!\!\text{ }\]
(6) If the integral of a function of the form \[\frac{{f}'(x)}{\sqrt{f(x)}}\] \[\int{\frac{{f}'\text{(}x\text{)}}{\sqrt{f\text{(}x\text{)}}}dx\,=2\sqrt{f\text{(}x\text{)}}+c}\]
(7) Standard substitutions
| Integrand form | Substitution | |
| (i) | \[\sqrt{{{a}^{2}}-{{x}^{2}}},\frac{1}{\sqrt{{{a}^{2}}-{{x}^{2}}}},{{a}^{2}}-{{x}^{2}}\] | \[x=a\sin \theta ,\] or \[x=a\cos \theta \] |
| (ii) | \[\sqrt{{{x}^{2}}+{{a}^{2}}},\,\frac{1}{\sqrt{{{x}^{2}}+{{a}^{2}}}},\,{{x}^{2}}+{{a}^{2}}\] | \[x=a\tan \theta \] or \[x=a\sin \text{h}\theta \] |
| (iii) | \[\sqrt{{{x}^{2}}-{{a}^{2}},}\,\,\frac{1}{\sqrt{{{x}^{2}}-{{a}^{2}}}},\,\,{{x}^{2}}-{{a}^{2}}\] | \[x=a\sec \theta \] or \[x=a\cosh \theta \] |
| (iv) | \[\sqrt{\frac{x}{a+x}},\,\,\sqrt{\frac{a+x}{x}},\,\,\sqrt{x(a+x)},\,\,\frac{1}{\sqrt{x(a+x)}}\] | \[x=a{{\tan }^{2}}\theta \] |
| (v) | \[\sqrt{\frac{x}{a-x}},\,\,\,\,\,\sqrt{\frac{a-x}{x}},\,\,\,\,\sqrt{x(a-x)},\,\,\frac{1}{\sqrt{x(a-x)}}\] | \[x=a{{\sin }^{2}}\theta \] |
| (vi) | \[\sqrt{\frac{x}{x-a}},\,\,\,\sqrt{\frac{x-a}{x}},\,\,\,\sqrt{x(x-a)},\,\,\,\frac{1}{\sqrt{x(x-a)}}\] | \[x=a{{\sec }^{2}}\theta \] |
| (vii) | \[\sqrt{\frac{a-x}{a+x}},\,\,\,\,\sqrt{\frac{a+x}{a-x}}\] | \[x=a\cos 2\theta \] |
| (viii) | \[\sqrt{\frac{x-\alpha }{\beta -x}},\,\,\,\sqrt{(x-\alpha )\,(\beta -x)},\,\,(\beta >\alpha )\] | \[x=\alpha {{\cos }^{2}}\theta +\beta {{\sin }^{2}}\theta \] |
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