Question

Find the indefinite integral

∫‍ (x)/sqrt(3-x) dx

Answer 1

Answer on Question #48247 – Math – Integral Calculus

Question.

Find the indefinite integral

$$\int \frac {x}{\sqrt {3 - x}} d x = ?$$

Solution.

$$\int {\frac {x}{\sqrt {3 - x}}} d x = \int {\frac {x - 3 + 3}{\sqrt {3 - x}}} d x = - \int {\sqrt {3 - x}} d x + \int {\frac {3}{\sqrt {3 - x}}} d x = = \int {\sqrt {3 - x}} d (3 - x) - 3 \int {\frac {d (3 - x)}{\sqrt {3 - x}}} =$$ $$\frac {2}{3} (3 - x) ^ {\frac {3}{2}} - 6 \sqrt {3 - x} + C = = \frac {2}{3} \sqrt {3 - x} (3 - x - 9) + C = - \frac {2}{3} (x + 6) \sqrt {3 - x} + C, \text{ where } C \text{ is an arbitrary real constant.}$$

Answer.

$$\int {\frac {x}{\sqrt {3 - x}}} d x = - \frac {2}{3} (x + 6) \sqrt {3 - x} + C$$

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