# Multi-method for solving definite integral

## Integration by parts

$$
\begin{align\*}
\ \\
\int f(x)g^\prime(x)dx &= f(x)g(x) - \int f^\prime(x)g(x) \cdot dx
\ \\
&\Downarrow
\ \\
\int\_a^b f(x)g^\prime(x)dx &= f(x)g(x)|\_a^b - \int\_a^b f^\prime(x)g(x) \cdot dx
\end{align\*}
$$

## Substitution method

$$
\begin{align\*}
&\int\_1^2 \frac{1}{\sqrt{5x-1}} \cdot dx
\ \\
\&t = \sqrt{5x-1}
\ \\
& t^2 = 5x-1
\ \\
&\frac{t^2 + 1}{5} = x
\ \\
\&d\[\frac{1}{5} (t^2 + 1)] = dx
\ \\
&\frac{1}{5} (2t) dt = dx
\ \\
& x | 1 \rightarrow 2
\\
& t | \sqrt{5 \cdot 1 -1} \rightarrow \sqrt{5 \cdot 2 -1}
\\
& t | 2 \rightarrow 3
\ \\
&\int\_2^3 \frac{1}{t} \cdot (\frac{1}{5} (2t) dt)
\ \\
&= \int\_2^3 \frac{2}{5} \cdot dt
\ \\
&= (\frac{2}{5} \cdot t)|\_2^3
\ \\
&= \frac{6}{5} - \frac{4}{5}
\ \\
&= \frac{2}{5}
\end{align\*}
$$