Question: "Use a table of integrals to evaluate the..."

hrostentsp6

Answered question

2021-11-07

Use a table of integrals to evaluate the following indefinite integral.

\int \frac{d x}{\sqrt{x^{2} - 6 x}}, x > 6

Sarythe

Beginner2021-11-08Added 11 answers

Step 1
To evaluate

\int \frac{d x}{\sqrt{x^{2} - 6 x}}

Step 2
Completing the square

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

= {(x - 3)}^{3} - {(3)}^{2}

Thus,

I = \int \frac{1}{\sqrt{x^{2} - 6 x}} d x

I = \int \frac{1}{\sqrt{{(x - 3)}^{2} - {(3)}^{2}}} d x

Step 3
Since,

\int \frac{1}{\sqrt{x^{2} - a^{2}}} d x = \ln | x + \sqrt{x^{2} - a^{2}} | + C

Thus,

I = \int \frac{1}{\sqrt{{(x - 3)}^{2} - {(3)}^{2}}} d x

I = \ln | (x - 3) + \sqrt{{(x - 3)}^{2} - {(3)}^{2}} | + C

I = \ln | (x - 3) + \sqrt{x^{2} - 6 x + 9 - 9} | + C

I = \ln | (x - 3) + \sqrt{x^{2} - 6 x} | + C

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