Indefinite Integrals
2.0 Methods of Integration
2.0 Methods of Integration
1. Integration by substitution (by change of independent variable)
We need to evaluate the integral of the form $$I = \int {f(g(x)).g'(x)dx} $$ in which $g(x)$ be the continuously differentiable function, we substitute $$g(x) = t \Rightarrow g'(x)dx = dt$$ in the integral such that $x$ will be replaced in terms of new variable $t$ i.e., $$I = \int {f(t)} dt$$ After evaluating the integral, the value of $t$ is substituted.
Question 2. Evaluate
(i) $\int {\left( {1 + {{\sin }^2}x} \right)\cos x} dx$
(ii) $\int {\frac{{{e^{n{{\tan }^{ - 1}}x}}}}{{1 + {x^2}}}dx} $
(iii) $\int {\frac{x}{{{x^4} + {x^2} + 1}}} dx$
Solution:
(i)$$ I = \int {\left( {1 + {{\sin }^2}x} \right)\cos x} dx $$Put $sin x = t \Rightarrow \cos xdx = dt$ $$ I = \int {(1 + {t^2})dt} $$ $$ I = \int {dt + \int {{t^2}dt} } $$ $$ I = t + \frac{{{t^3}}}{3} + C $$Put the value of $t$ $$ I = \sin x + \frac{{{{\sin }^3}x}}{3} + C $$
(ii) $$ I = \int {\frac{{{e^{n{{\tan }^{ - 1}}x}}}}{{1 + {x^2}}}dx} $$Put $n{\tan ^{ - 1}}x = t \Rightarrow \frac{n}{{1 + {x^2}}}dx = dt \Rightarrow \frac{{dx}}{{1 + {x^2}}} = \frac{{dt}}{n}$ $$ I = \int {{e^t}} .\frac{{dt}}{n} $$ $$ I = \frac{1}{n}\int {{e^t}} dt $$$$ I = \frac{{{e^t}}}{n} + C $$Put the value of $t$ $$ I = \frac{{{e^{n{{\tan }^{ - 1}}x}}}}{n} + C $$
iii) $$ I = \int {\frac{x}{{{x^4} + {x^2} + 1}}} dx $$ $$ I = \int {\frac{x}{{{{\left( {{x^2}} \right)}^2} + {x^2} + 1}}} dx $$Put,$${x^2} = t\;\; \Rightarrow 2xdx = dt\;\; \Rightarrow xdx = \frac{{dt}}{2} $$ $$ I = \frac{1}{2}\int {\frac{{dt}}{{{t^2} + t + 1}}} $$ $$ I = \frac{1}{2}\int {\frac{{dt}}{{{{\left( {t + \frac{1}{2}} \right)}^2} + {{\left( {\frac{{\sqrt 3 }}{2}} \right)}^2}}}} $$ $$ I = \frac{1}{2}.\frac{1}{{\sqrt 3 /2}}{\tan ^{ - 1}}\left( {\frac{{t + \frac{1}{2}}}{{\frac{{\sqrt 3 }}{2}}}} \right) + C $$ $$ I = \frac{1}{{\sqrt 3 }}{\tan ^{ - 1}}\left( {\frac{{2t + 1}}{{\sqrt 3 }}} \right) + C $$Put the value of $t$, $$I = \frac{1}{{\sqrt 3 }}{\tan ^{ - 1}}\left( {\frac{{2{x^2} + 1}}{{\sqrt 3 }}} \right) + C $$
