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Question Number 168473 by Mastermind last updated on 11/Apr/22

$$Integrate:$$$$\int {(\frac{1}{x}-1)}^{\frac{1}{2}}dx$$$$$$$$Mastermind$$

Answered by MJS_new last updated on 11/Apr/22

$$\int \sqrt{\frac{1-x}{x}}dx=$$$$[t=\sqrt{\frac{1-x}{x}}\to dx=-2x^{2}tdt]$$$$=-2\int \frac{t^2}{{(t^2+1)}^2}dt=\frac{t}{t^2+1}-\arctan t=$$$$=\sqrt{(1-x)x}-\arctan \sqrt{\frac{1-x}{x}}+C$$

Commented by Mastermind last updated on 12/Apr/22

$$$$$$Thanksman$$

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