techniques of integration
Techniques of integration
Let \[F(x)\] be an antiderivative of \[f(x)\] and \[G(x)\] be an antiderivative of \[g(x)\].
As a rule of thumb, always try to simply a function and integrate using known functions, then try the substitution method and finally integrate it by parts.
Integral of power
\[\int x^{n}\,dx=\frac{x^{n+1}}{n+1}+C\] and \[\int_{0}^{b}x^{n}\,dx=\frac{b^{n+1}}{n+1}\].
Integration by substitution
\[\int f(g(x))\cdot g^{\prime}(x)\,dx=F(g(x))+C\] and \[\int_{a}^{b} f(g(x))\cdot g^{\prime}(x)\,dx=\int_{g(a)}^{g(b)}f(u)\,du\].
Integration by parts
Conventionally written as \[\int u(x)v^{\prime}(x)\,dx=u(x)v(x)-\int u^{\prime}(x)v(x)\,dx\] and \[\int_{a}^{b}u(x)v^{\prime}(x)=\biggl[ u(x)v(x) \biggr]_{a}^{b}-\int_{a}^{b}u^{\prime}(x)v(x)\,dx\].
Another notation that can be used for ease of understanding would be \[\int f(x)g(x)\,dx=f(x)G(x)-\int f^{\prime}(x)G(x)\,dx\].
Integral of inverse functions
Let \[y=f(x)\] then \[x=f^{-1}(y)\]. Then, \[\int f^{-1}(y)\,dy=xf(x)-\int f(x)\,dx\] or sometimes written as \[\int f^{-1}(y)\,dy=yf^{-1}(y)-\left( F\circ f^{-1} \right)(y)\].