Given that the function f (x) satisfies 2F (x) - f (1 / x) = 3 / X & # 178;, find the expression of F (x)

Given that the function f (x) satisfies 2F (x) - f (1 / x) = 3 / X & # 178;, find the expression of F (x)

2f(x)-f(1/x)=3/x²
Replace x with 1 / X to get
2f(1/x)-f(x)=3x²
Then f (x) and f (1 / x) are treated as two variables to solve the equation
obtain
f(x)=(2/x^2)+x^2

Indefinite integral of 2x / (1 + X & # 178; + x)

∫ [2x/(x^2+x+1) ]dx= ∫ [(2x+1)/(x^2+x+1) ]dx -∫ dx/(x^2+x+1) =ln|x^2+x+1| -∫ dx/(x^2+x+1) considerx^2+x+1 = (x+1/2)^2 + 3/4letx+1/2 = (√3/2)tanydx = (√3/2)(secy)^2 dy∫dx/(x^2+x+1) =(2√3/3)∫ d...

Seeking indefinite integral ∫ 1 / (1-2x-x & # 178;) ^ 1 / 2DX

We will be the first (1-2x-x-\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\2) from the basic integral formula

How to prove the second derivative of inverse function d (1 / y ') / DX = - y' '/ (y') ^ 2
Such as the title

The original formula is: (- Y "/ y '^ 2) DX / DX = - y) / y' ^ 2

(1 / y ')'Is equal to what tut online mathematical experts! Inverse function derivation!
QAQ as above

(1/y')'=-y″/(y′)²

It is known that the derivative y ', y' of y = y (x) exists, and the inverse function of y = y (x) is x = x (y). Try y ', y' to express (d ^ 2x) / DX ^ 2
—（y''）/(y')^3

x=x(y)
x'=dx/dy=1/(dy/dx)=1/y'(x)
x''=d(1/y')/dy=d(1/y')/dx*(dx/dy)
x"=(-1/(y'）^2)*y"*(1/y')=-y"/(y')^3

Let sin2x be an original function of F (x) to find f (x)

Classmate, what you want is: ∫ f (x) DX
Sin2x is a primitive function of F (x), so ∫ f (x) DX = sin2x + C definite integral is to find the set of primitive functions, ∫ f (x) DX means to find the set of primitive functions of F (x). Sin2x is a primitive function of F (x). Adding sin2x to any constant C is the primitive function of F (x)

How to solve this problem

To find the answer of indefinite integral ∫ xex power DX, we need to solve the problem process. This is a calculation problem, and it needs to be solved step by step

∫xe^xdx
=∫xde^x
=xe^x-∫e^xdx
=xe^x-e^x+C

Seeking indefinite integral ∫ (LNX) &# 178; DX

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