SiNx function definite integral Logically, after the definite integral is - cosx, and the area on [0, pie] is 2 But the average value of a period of sine function can be regarded as (SiNx) max / radical 2 It makes me very tangled. In this case, the area should be root 2 * Pie / 2 Then I was forced to get the pie = 2 root sign. 2. There seems to be a problem, The above should be [0, pie], half a cycle, or consider according to | SiNx |

SiNx function definite integral Logically, after the definite integral is - cosx, and the area on [0, pie] is 2 But the average value of a period of sine function can be regarded as (SiNx) max / radical 2 It makes me very tangled. In this case, the area should be root 2 * Pie / 2 Then I was forced to get the pie = 2 root sign. 2. There seems to be a problem, The above should be [0, pie], half a cycle, or consider according to | SiNx |

The average value of SiNx is 0

Find the indefinite integral of function y = SiNx / (1 + SiNx)

∫[sinx/(1+sinx)]dx
=∫dx-∫[1/(1+sinx)]dx
=∫dx-∫{1/[1+cos(π/2-x)]}dx
=∫dx-1/2∫{1/[cos(π/4-x/2)]^2}dx
=x+tan(π/4-x/2)+C

Why is the definite integral of X to the nth power on (0,1) 1 / N + 1?

The definite integral of the nth power of X on (0,1) = the difference after 1 / (n + 1) * x ^ (n + 1) is substituted into 1 and 0,
That is, 1 / (n + 1) * 1 ^ (n + 1) - 1 / (n + 1) * 0 ^ (n + 1) = 1 / (n + 1)

How to find the definite integral of the (- x) power of E from negative infinity to 0 The specific question is as follows: when x is greater than or equal to 0, find f (x) = (1 / 2) [∫ e ^ (- x) DX (the lower integral limit is negative infinity and the upper limit is 0)] + (1 / 2) [∫ e ^ (- x) DX (the lower integral limit is 0 and the upper limit is x)]. How is the answer 1 - (1 / 2) e ^ (- x)

The definite integral of the (- x) power of E from negative infinity to 0 is - 1 / 2 + 1 / 2 * e (infinite power), that is, positive infinity
From the answer, the original function should be:
F (x) = (1 / 2) [∫ e ^ (x) DX (integral lower limit is negative infinity, upper limit is 0)] + (1 / 2) [∫ e ^ (- x) DX (integral lower limit is 0, upper limit is x)]

What is the total differential DZ of the function z = x squared + 2XY squared + 4Y to the third power

Solution;
z(x)=2x+2y ²
z(y)=4xy+12y ²
dz=(2x+2y ²) dx+(4xy+12y ²) dy

Find the total differential of the binary function z = e ^ XY at point (1,2)

Z=e^xy
The derivative at x is Ye ^ (XY)
The derivative at Y is Xe ^ (XY)
dz=ye^(xy)dx+xe^(xy)dy
=2e^2dx+e^2dy