Step by step solution to "Find the following indefinite integral: ∫12x(x2−9)−3"

High School
Calculus and Analysis
Calculus 2
Applications of integrals

guringpw

Answered question

2021-12-09

Find the following indefinite integral:

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

Answer & Explanation

hysgubwyri3

Beginner2021-12-10Added 43 answers

Step 1
Given: $I = \int 12 x {(x^{2} - 9)}^{- 3} d x$
for evaluating given integral, we substitute
$x^{2} - 9 = t$ ...(1)
now differentiating equation (1) with respect to x
so,
$\frac{d}{d x} (x^{2} - 9) = \frac{d}{d x} (t)$
$\frac{d}{d x} (x^{2}) - \frac{d}{d x} (9) = \frac{d t}{d x} (∵ \frac{d}{d x} (x^{n}) = n x^{n - 1})$
$2 x - 0 = \frac{d t}{d x}$
2xdx=dt
Step 2
now, replacing 2xdx with dt, $(x^{2} - 9)$ with t in given integral
so,
$\int 12 x {(x^{2} - 9)}^{- 3} d x = 6 \int t^{- 3} d t (∵ \int t^{n} d t = \frac{t^{n + 1}}{n + 1} + c)$
$= 6 (\frac{t^{- 2}}{- 2}) + c$
$= - \frac{3}{t^{2}} + c$
now, substituting $t = x^{2} - 9$ in equation (2)
$\int 12 x {(x^{2} - 9)}^{- 3} d x = - \frac{3}{{(x^{2} - 9)}^{2}} + c$
hence, given integral is equal to $- \frac{3}{{(x^{2} - 9)}^{2}} + c$ .

Ethan Sanders

Beginner2021-12-11Added 35 answers

$\int 12 x \cdot {(x^{2} - 9)}^{- 3} d x$
$12 \cdot \int x \cdot {(x^{2} - 9)}^{- 3} d x$
$12 * \int \frac{1}{2 t^{3}} d t 12 * \frac{1}{2} * \int \frac{1}{t^{3}} d t$
Calculate integral
$6 \cdot (- \frac{1}{2 t^{2}})$
Substitute back
$6 \cdot (- \frac{1}{2 {(x^{2} - 9)}^{2}})$
Simplify
$- \frac{3}{{(x^{2} - 9)}^{2}}$
Answer:
$- \frac{3}{{(x^{2} - 9)}^{2}} + C$

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