Key to "Evaluate the integrals ∫dxx(9−x2)"

High School
Calculus and Analysis
Calculus 2
Applications of integrals

Kathy Williams

Answered question

2021-12-13

Evaluate the integrals

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

Answer & Explanation

temzej9

Beginner2021-12-14Added 30 answers

Step 1
Given Data
The integral is $\int \frac{d x}{x} (9 - x^{2})$
Simplify the given integral,
$\int \frac{d x}{x (9 - x^{2})} = - \int \frac{d x}{x (x^{2} - 9)}$
$= - \int \frac{d x}{x^{3} (1 - \frac{9}{x^{2}})}$
Step 2
Let,
$1 - \frac{9}{x^{2}} = t$
$\frac{d}{d x} (1 - \frac{9}{x^{2}}) = \frac{d}{d x} (t)$
$\frac{18}{x^{3}} = \frac{d t}{d x}$
$d x = \frac{x^{3}}{18} d t$
Step 3
Put the values and solve the given integral,
$\int \frac{d x}{x (9 - x^{2})} = - \int \frac{(\frac{x^{3}}{15}) d t}{x^{3} t}$
$= - \frac{1}{18} \int \frac{d t}{t}$
$= - \frac{1}{18} \ln t + C$
$= - \frac{1}{18} \ln (1 - \frac{9}{x^{2}}) + C$
$= - \frac{1}{18} \ln (\frac{x^{2} - 9}{x^{2}}) + C$
Hence, the given integral is $- \frac{1}{18} \ln (\frac{x^{2} - 9}{x^{2}}) + C$

SlabydouluS62

Skilled2021-12-15Added 52 answers

Given:

\int \frac{1}{x \cdot (9 - x^{2})} d x

Rewrite the fraction

\int \frac{1}{9 x} + \frac{x}{9 (9 - x^{2})} d x

\int \frac{1}{9 x} d x + \int \frac{x}{9 (9 - x^{2})} d x

Calculate

\frac{1}{9} \cdot \ln (| x |) - \frac{1}{18} \cdot \ln (| 9 - x^{2} |)

Result:

\frac{1}{9} \cdot \ln (| x |) - \frac{1}{18} \cdot \ln (| 9 - x^{2} |) + C

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