Expert Answer to "The integral: dx(4x2−24x+27)32"

Harlen Pritchard

Answered question

2020-11-23

The integral:

\frac{d x}{{(4 x^{2} - 24 x + 27)}^{\frac{3}{2}}}

brawnyN

Skilled2020-11-24Added 91 answers

4 x^{2} - 24 x + 27 = {(2 x - 6)}^{2} - 3^{2}

let 2x-6 = t then 2dx = dt
so,

\int \frac{1}{{(4 x^{2} - 24 x + 27)}^{\frac{3}{2}}} d x = \frac{1}{2} \int \frac{1}{{(t^{2} - 3^{2})}^{\frac{3}{2}}} d t

now we use the integration formula

\int \frac{1}{{(u^{2} - a^{2})}^{\frac{3}{2}}} d u = \frac{- u}{a^{2} \sqrt{u^{2} - a^{2}}} + C

\frac{1}{2} \int \frac{1}{{(t^{2} - 3^{2})}^{\frac{3}{2}}} d t = \frac{- t}{18 \sqrt{t^{2} - 9}} + C

Then,

\int \frac{1}{{(4 x^{2} - 24 x + 27)}^{\frac{3}{2}}} d x = \frac{6 - 2 x}{18 \sqrt{4 x^{2} - 24 x + 27} + C} = \frac{3 - x}{9 \sqrt{4 x^{2} - 24 x + 27}} + C

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?