Задача 80936 вычислите определенные интегралы...

6a07ec291a12cf5d38d7c6c5

5/16/2026 4:01:59 AM

вычислите определенные интегралы

626fab9a354ab65fb2f43abb

5/16/2026 11:07:47 AM

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1) [m]\int \limits_{-1}^4 (x^2 - 5x) dx = (\frac{x^3}{3} - \frac{5x^2}{2}) \bigg |_{-1}^4 = (\frac{4^3}{3} - \frac{5 \cdot 4^2}{2}) - (\frac{(-1)^3}{3} - \frac{5(-1)^2}{2}) = [/m]
[m]= \frac{64}{3} - \frac{5 \cdot 16}{2} - (-\frac{1}{3}) + \frac{5}{2} = 21 \frac{2}{3} - 40 + 2 \frac{1}{2} = -17 + \frac{4}{6} + \frac{3}{6} = -17 + \frac{7}{6} = -15 \frac{5}{6}[/m]

2) [m]\int \limits_1^3 (4x - x^3) dx = (\frac{4x^2}{2} - \frac{x^4}{4}) \bigg |_1^3 = (2 \cdot 3^2 - \frac{3^4}{4}) - (2 \cdot 1^2 - \frac{1^4}{4}) = [/m]
[m]= 18 - \frac{81}{4} - 2 + \frac{1}{4} = 16 - \frac{80}{4} = 16 - 20 = -4[/m]

3) [m]\int \limits_0^2 (3x^3 + 5x - 8) dx = (\frac{3x^4}{4} + \frac{5x^2}{2} - 8x) \bigg |_0^2 = (\frac{3 \cdot 2^4}{4} + \frac{5 \cdot 2^2}{2} - 8 \cdot 2) - (\frac{3 \cdot 0^4}{4} + \frac{5 \cdot 0^2}{2} - 0) = [/m]
[m]= \frac{3 \cdot 16}{4} + \frac{5 \cdot 4}{2} - 16 = 12 + 10 - 16 = 6[/m]

4) [m]\int \limits_0^{\pi/4} 2 \sin x dx = -2 \cos x \bigg |_0^{\pi/4} = -2 \cos \frac{\pi}{4} - (-2 \cos 0) =[/m]
[m]= -2 \cdot \frac{\sqrt{2}}{2} + 2 \cdot 1 = -\sqrt{2} + 2 = 2 - \sqrt{2}[/m]