2021 HAPPY NEW Year 1)∫((x^3 +3x+2)/((x^2 +1)^2 (x+1)))dx 2)∫((2cos(x)−sin(x))/(3sin(x)+5cos(x)))dx 3)∫((tan(2x))/( (√(sin^6 (x)+cos^6 (x)))))dx 4)∫x(√((1−x^2 )/(1+x^2 ))) dx


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Question Number 127870 by Eric002 last updated on 02/Jan/21

$2021$ $H A P P Y N E W Y e a r$ $1) \int \frac{x^{3} + 3 x + 2}{{(x^{2} + 1)}^{2} (x + 1)} d x$ $2) \int \frac{2 c o s (x) - s i n (x)}{3 s i n (x) + 5 c o s (x)} d x$ $3) \int \frac{t a n (2 x)}{\sqrt{{s i n}^{6} (x) + {c o s}^{6} (x)}} d x$ $4) \int x \sqrt{\frac{1 - x^{2}}{1 + x^{2}}} d x$
	Answered by Dwaipayan Shikari last updated on 02/Jan/21

	$\int x \sqrt{\frac{1 - x^{2}}{1 + x^{2}}} d x x^{2} = u \Rightarrow 2 x = \frac{d u}{d x}$ $= \frac{1}{2} \int \sqrt{\frac{1 - u}{1 + u}} = \frac{1}{2} \int \frac{1}{\sqrt{1 - u^{2}}} - \frac{u}{2 \sqrt{1 - u^{2}}} d u$ $= \frac{1}{2} {s i n}^{- 1} (x^{2}) + \frac{1}{2} \sqrt{1 - x^{4}}$
	Commented by Eric002 last updated on 02/Jan/21

	$t h a n k y o u$
	Answered by liberty last updated on 03/Jan/21

	$(2) \int \frac{- \sin x + 2 \cos x}{3 \sin x + 5 \cos x} dx =$ $\frac{- \sin x + 2 \cos x}{3 \sin x + 5 \cos x} = P (\frac{3 \sin x + 5 \cos x}{3 \sin x + 5 \cos x}) + Q (\frac{3 \cos x - 5 \sin x}{3 \sin x + 5 \cos x})$ $\Rightarrow - \sin x + 2 \cos x = (3 P - 5 Q) \sin x + (5 P + 3 Q) \cos x$ $(\begin{matrix} 3 - 5 \\ 5 3 \end{matrix}) (\begin{matrix} P \\ Q \end{matrix}) = (\begin{matrix} - 1 \\ 2 \end{matrix}) {\begin{cases} P = \frac{\| \begin{array}{c} - 1 - 5 \\ 2 3 \end{array} \|}{34} = \frac{7}{34} \\ Q = \frac{\| \begin{array}{c} 3 - 1 \\ 5 2 \end{array} \|}{34} = \frac{11}{34} \end{cases}$ $then \int \frac{- \sin x + 2 \cos x}{3 \sin x + 5 \cos x} dx = \frac{7 x}{34} + \frac{11}{34} ℓ n ∣ 3 \sin x + 5 \cos x ∣ + C$
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