"Find the following indefinite integral: ∫12x(x2−9)−3dx" was answered by our community

a2linetagadaW

Answered question

2021-11-07

Find the following indefinite integral:

\int 12 x {(x^{2} - 9)}^{- 3} d x

pierretteA

Skilled2021-11-08Added 102 answers

Step 1
The given integral is

\int 12 x {(x^{2} - 9)}^{- 3} d x

.
Let

u = x^{2} - 9

.
This implies that du=2xdx.
Substitute these values in the above integral and simplify as follows.
Step 2

\int 12 x {(x^{2} - 9)}^{- 3} d x = 6 \int 2 x {(x^{2} - 9)}^{- 3} d x

= 6 \int u^{- 3} d u

= 6 \frac{u^{- 3 + 1}}{- 3 + 1} + C

= \frac{6}{- 2} u^{- 2} + C

= - 3 u^{- 2} + C

= - 3 {(x^{2} - 9)}^{- 2} + C

Thus,

\int 12 x {(x^{2} - 9)}^{- 3} d x = - 3 {(x^{2} - 9)}^{- 2} + C

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