Question

大学数学题目,请数学高手作答,非常感激. 1、求导数： 函数,不定积分

题目如图片
放大之后把图片另存为再打开就可以看清楚了,非常感谢!

Answer 1

(1)：y = x²sin2x
y' = sin2x • 2x + x² • 2cos2x
= 2xsin2x + 2x²cos2x
= 2x(sin2x + xcos2x)
(2)：∫ xlnx dx = ∫ lnx d(x²/2)
= (x²/2)lnx - (1/2)∫ x² d(lnx)
= (1/2)x²lnx - (1/2)∫ x² • 1/x dx
= (1/2)x²lnx - (1/2)∫ x dx
= (1/2)x²lnx - (1/2)(x²/2) + C
= (x²/4)(2lnx - 1) + C
(3)：∫ (2 - 3x)¹² dx
令u = 2 - 3x，du = (2)'dx - (3x)'dx = -3dx
原式= ∫ u¹² • du/(-3)
= (-1/3) • u¹³/13 + C
= (-1/39)(2 - 3x)¹³ + C
(4)：
f(x) = 3x³ - 9x + 7
f'(x) = 9x² - 9
f''(x) = 18x
f'(x) = 0 => x = ±1
f''(-1) < 0，取得极大值，f''(1) > 0，取得极小值
极大值f(-1) = 13，极小值f(1) = 1
在端点，f(-3) = -47，f(3) = 61
故在x∈[-3，3]中，最大值为61，最小值为-47

追问

非常感谢你的帮忙，请再帮忙做几道题，实太太感谢你了！

追答

(1)：y = cosx，y' = - sinx，y'' = - cosx，y''' = sinx
(2)：∫ (2^x + x²) = ∫ 2^x dx + ∫ x² dx = (2^x)/ln2 + x³/3 + C
(3)：y = ax^(b + c)
y' = a(b + c)x^(b + c - 1)
(4)：
lim(x→2) (x² - x + k)/(x - 2) = 3，分子能够因式分解，并且能约去x - 2项
所以(x² - x + k) = (x - 2)(x + m)
x² - x + k = x² + (m - 2)x - 2m
{ m - 2 = -1 => m = 1
{ k = - 2m => k = (-2)(1) = -2
所以k = -2
(5)：∫ dx/(3 + 2x) = (1/2)∫ d(2x)/(3 + 2x) = (1/2)∫ d(3 + 2x)/(3 + 2x) = (1/2)ln|3 + 2x| + C
(6)：d(？) = (2^x)ln2 dx
d(？)/dx = (2^x)ln2
？= (ln2)∫ 2^x dx = (ln2) • (2^x)/ln2 + C = 2^x + C
故d(2^x + C) = (2^x)ln2 dx，C为任意的常数

追问

实在太感谢你了！还有到你几道题能再麻烦你吗?希望能得到你的帮助,谢谢!

追答

(1)：∫ lnx/x³ dx
= ∫ lnx d[-1/(2x²)]
= - lnx/2x² + (1/2)∫ x² d(lnx)
= -lnx/2x² + (1/2)∫ x dx
= -lnx/2x² + x²/4 + C
(2)：y = 1/x，y' = -1/x²，y(4) = 1/4
y' = -1/16
y = (-1/16)(x - 4) + 1/4
y = (-1/16)x + 1/2
(3)：y = x³ - 3x² - 9x + 10
y' = 3x² - 6x - 9 = 3(x² - 2x - 3)
y'' = 6x - 6 = 6(x - 1)
y' = 0 => x = 3 或 x = -1
y''(-1) 0，取得极小值
极大值f(-1) = 15，极小值f(3) = -17
在端点：f(-2) = 8，f(6) = 64
所以在[-2，6]的最大值是64，最小值是-17
(4)：y = 2 + x - x²，抛物线开口向下
y' = -2x + 1
y' = 0 => x = 1/2
所以极大值f(1/2) = 9/4

Answer 2

1、求导数
导数=2x sin2x+2x^2 con2x
2.不定积分=(1/2) (ln x)^2 + c
3.t=3x-2,
(2-3x )^12= )(3x-2)^12
dx=dt/3 ,
不定积分=(1/39)(3x-2)^13+c
4
F=3X [X+(3)^0.5] [X-(3)^0.5] +7

x=3. Fmax=54+7=61,
x=-3, Fmin= -54+7=-47

追问

实在太感谢你了,我还有几道题能再麻烦你吗?谢谢

Answer 3

what啊

Answer 4

1:等号两边都Lg,,,,