Explained: "Find integral ∫x2−9xdx"

Reeves

Answered question

2021-10-15

Find integral

\int \frac{\sqrt{x^{2} - 9}}{x} d x

Margot Mill

Skilled2021-10-16Added 106 answers

Step 1
Consider the integrals.

\int \frac{\sqrt{x^{2} - 9}}{x} d x

Let,

x = 3 \sec (t)

Differentiating with respect to "t"

d x = 3 \sec (t) \cdot \tan (t) d t

Step 2
Substituting all value in given integrals,

\int \frac{\sqrt{x^{2} - 9}}{x} d x = \int \frac{\sqrt{9 \sec^{2} t - 9}}{3 \sec (t)} 3 \sec (t) \cdot \tan (t) d t

= \int \frac{3 \sqrt{\sec^{2} t - 1}}{3 \sec (t)} 3 \sec (t) \cdot \tan (t) d t

= \int 3 \cdot \tan (t) \cdot \tan (t) d t

= 3 \int \tan^{2} (t) d t

Further solving it,

= 3 \int (\sec^{2} (t) - 1) d t ∵ \sec^{2} (t) - 1 = \tan^{2} (t)

= 3 (\tan t - t) + C

= 3 [\tan (\sec^{- 1} (\frac{x}{3})) - \sec^{- 1} (\frac{x}{3})] + C

= - 3 \sec^{- 1} (\frac{x}{3}) + \sqrt{x^{2} - 9} + C

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