całka maks: Oblicz ∫₀^π/2 cosx*ln(sin³x + cos³x)dx

Mariusz: ∫₀^π/2cos(x)ln(sin³x+cos³x)dx Nieoznaczoną przez części ∫cos(x)ln(sin³x+cos³x)dx=sin(x)ln(sin³x+cos³x)−

	(3sin2xcosx−3cos2xsinx)sinx
∫		dx
	sin3x+cos3x

Mariusz: ∫cos(x)ln(sin³(x)−cos³(x))dx=sin(x)ln(sin³(x)+cos³(x))−

	sin3xcosx−cos2(x)sin2(x)
3∫		dx
	sin3(x)+cos3(x)

	sin3xcosx−cos2(x)sin2(x)
∫		dx
	sin3(x)+cos3(x)

cos(x)=(1−sin(x))t cos²(x)=(1−sin(x))²t² 1−sin²(x)=(1−sin(x))²t² (1+sin(x))(1−sin(x))=(1−sin(x))²t² (1−sin(x))(1+sin(x)−(1−sin(x))t²)=0 1+sin(x)−(1−sin(x))t²=0 1+sin(x)−t²+sin²(x)t²=0 sin(x)(t²+1)=t²−1

	t2−1
sin(x)=
	t2+1

	t2+1−t2+1
cos(x)=(		)*t
	t2+1

	2t
cos(x)=
	t2+1

	t2−1
sin(x)=
	t2+1

	2t(t2+1)−2t(t2−1)
cos(x)dx=		dt
	(t2+1)2

	2t	2
cos(x)dx=			dt
	t2+1	t2+1

2t		2t	2
	dx=			dt
t2+1		t2+1	t2+1

	2
dx=		dt
	t2+1

	sin3xcosx−cos2(x)sin2(x)
∫		dx
	sin3(x)+cos3(x)

	2t(t2−1)3−4t2(t2−1)2	(t2+1)3	2
∫				dt
	(t2+1)4	(t2−1)3+(2t)3	t2+1

	(t2−1)2(t3−2t2−t)
4∫
	(t2+2t−1)(t4−2t2+1−2t3+2t+4t2)(t2+1)2

	(t2−1)2(t3−2t2−t)
4∫		dt
	(t2+1)2(t2+2t−1)(t4−2t3+2t2+2t+1)

t⁴−2t³+2t²+2t+1 (t⁴−2t³)−(−2t²−2t−1) (t⁴−2t³+t²)−(−t²−2t−1) (t²−t)²−(−t²−2t−1)

	y		y2
(t2−t+		)2−((y−1)t2+(−y−2)t+		−1)
	2		4

	y2
4(		−1)(y−1)−(y+2)2=0
	4

(y²−4)(y−1)−(y+2)²=0 (y−2)(y+2)(y−1)−(y+2)²=0 (y+2)(y²−3y+2−y−2)=0 y(y+2)(y−4)=0

	y		y2
(t2−t+		)2−((y−1)t2+(−y−2)t+		−1)
	2		4

(t²−t+2)²−(3t² −6t+3) (t²−t+2)²−(√3t−√3)² (t²−(1−√3)t+2−√3)(t²−(1+√3)t+2+√3)

	(t2−1)2(t3−2t2−t)
4∫		dt
	(t2+1)2(t2+2t−1)(t2−(1−√3)t+2−√3)(t2−(1+√3)t+2+√3)

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