Property 13
Partial Integration
\((u\cdot v)'=u'\cdot v+v'\cdot u\)
\(d(u\cdot v)=vdu+udv\)
\(udv=d(u\cdot v)-vdu\)
\(\displaystyle\int udv=\displaystyle\int d(u\cdot v)-\displaystyle\int vdu\)
\(\boxed{\displaystyle\int udv=u\cdot v-\displaystyle\int vdu}\)
The selection of u and dv in the partial integration is very important. For this selection the below ordering are to be taken into consideration. As going down taking integral gets easier.
\(\begin{array}{|c|l|}\hline L&\text{Logarithm}\\\hline A&\text{Arc (Inverse Trigonometrical Function)}\\\hline P&\text{Polynomial}\\\hline T&\text{Trigonometrical Function (sin, cos, tan, cot)}\\\hline\text{Ü}&\text{Exponential Function }(a^x)\\\hline\end{array}\)
Example
\(\displaystyle\int x\cdot e^x\,dx=?\)
Answer
\(\displaystyle\int x\cdot e^x\,dx\)
\(\begin{aligned}x&\to\text{polynomial}\\e^x&\to\text{exponential function}\end{aligned}\)
\(\begin{aligned}x&=u& e^x dx&=dv\\dx&=du&e^x&=v\end{aligned}\)
\(\begin{aligned}\displaystyle\int xe^x\,dx&=\displaystyle\int udv\\&=u\cdot v-\displaystyle\int vdu\\&=x\cdot e^x-\displaystyle\int e^x\,dx\\&=x\cdot e^x-e^x+c\end{aligned}\)
1.
\(\displaystyle\int x\cdot\sin x\,dx=?\)
2.
\(\displaystyle\int x^2\cdot e^x\,dx=?\)
3.
\(\displaystyle\int \ln x\,dx=?\)
4.
\(\displaystyle\int \arccos x\,dx=?\)
5.
\(\displaystyle\int \arctan x\,dx=?\)