Is the Tabular Method the Easiest Way to Solve Integrals?

[karush]

$\textsf{1) Evaluate the integral}$
\begin{align*}
I_1&=\int x \csc^2 6x \, dx\\
u&=6x\therefore du=dx\\
dv&=\csc^2(6x) \, dx \therefore v=-\frac{1}{6}\cot(6x) \\
u&=6x\therefore du=6 \, dx \\
I&=-\frac{1}{6} x\cot(6x)+\frac{1}{36}\int \cot(u) \, dx\\
&=\color{red}{
-\frac{1}{6} x\cot(6x)
+\frac{1}{36}\ln(|\sin36)|)}
\end{align*}

Ok I think this is correct, if so
saw another student use the tabular method to solve this
but couldn't see good enought to understand it
is that a lot easier.

 

[MarkFL]

I think you've gotten your $u$'s mixed up. We are given:

\(\displaystyle I=\int x\csc^2(6x)\,dx\)

Let:

\(\displaystyle u=x\implies du=dx\)

\(\displaystyle dv=\csc^2(6x)\,dx\implies v=-\frac{1}{6}\cot(6x)\)

Hence:

\(\displaystyle I=-\frac{x}{6}\cot(6x)+\frac{1}{6}\int \cot(6x)\,dx=\frac{1}{36}\ln\left|\sin(6x)\right|-\frac{x}{6}\cot(6x)+C\)

I don't know anything about the "tabular method." :D

 

[skeeter]

tabular integration ...

$u$ and its derivatives in the second column

$dv$ and its antiderivatives in the third column

View attachment 6450

 

[karush]

well that is certainly much more usefull

I guess the uv method was to illustrate how it works short of a full proof

like your chart!

 

[Ackbach]

This method is also called the "Stand and Deliver" method, because it was featured in that movie.