Hi there,
I've been given the problem attached and its causing me lots of problems. When I perform the substitution I don't know how to "correct" du so that the new integral has a denominator that is equivalent to t-2.
Please help!
Hi there,
I've been given the problem attached and its causing me lots of problems. When I perform the substitution I don't know how to "correct" du so that the new integral has a denominator that is equivalent to t-2.
Please help!
$\displaystyle \displaystyle \begin{align*} \int{ \frac{ \sqrt{ t - 1 } }{ t - 2 } \, dt } &= 2 \int{ \frac{ t - 1 }{ 2 \, \sqrt{ t - 1} \left( t - 2 \right) } \, dt } \end{align*}$
Now make the substitution $\displaystyle \displaystyle \begin{align*} u = \sqrt{ t - 1} \implies du = \frac{1}{2 \, \sqrt{t - 1} } \, dt \end{align*}$ and the integral becomes
$\displaystyle \displaystyle \begin{align*} 2 \int{ \frac{ u^2 }{ u^2 - 1 } \, du } &= 2 \int{ 1 + \frac{1}{u^2 - 1} \, du} \end{align*}$
And now you need to apply Partial Fractions to continue.