How to find the integral of a composite function?

The integration of a composite function can generally be solved using the commutation method. Not only the integration variable will change with it, but also the integration limit will change with it. For example:

Extended information: If the function y=f(u) defines the domain.

is b, u=g(x) is a, then the composite function y=f[g(x)] is d= consider the range of x values of each part, and take their intersection.

Finding the definition domain of a function should mainly consider the following points:

When it is an integer. or odd roots, the range of r.

When it is an even radical, the number of squares to be opened is not less than 0 (i.e., 0);

When it is a fraction. , the denominator is not 0; When the denominator is an even radical, the number of squares to be opened is greater than 0;

When exponential, the base is not 0 for the exponential power of zero or negative integer exponential power (e.g., medium).

When it is formed by combining some basic functions through four operations, its definition domain should be the set of values of independent variables that make each part meaningful, that is, find the intersection of the set of definition domains of each part.

The definition domain of a piecewise function is the union of the set of values of the independent variables on each segment.

Functions built by practical problems should consider not only the requirements of the arguments for the analytic expression, but also the requirements of the arguments for the practical meaning.

For functions with parameter letters, the values of the letters should be classified and discussed when finding the definition domain, and it should be noted that the definition domain of the function is a non-empty set.

Logarithmic functions. The true number of must be greater than zero, and the base number must be greater than zero and not equal to 1.

Trigonometric function. The cutting function in should be aware of the constraints of the diagonal variable.

Original = integral (1, -1) (-1 2 root number (1-x 2) d (1-x 2) = 0

2.Primitive = integral(0, infinite) -xde (-x)=-e (-x)(0, infinity) = 1

3.Primitive = integral (0, infinite) (t+5) e (-t) dt = integral (0, infinite) te (-t) dt + integral (0, infinity) 5e (-t) dt = 1+5e (-t) (0, infinity) = 1-5 = -4

I don't know what you mean by integral compound function integral.

It's that type, but there are two ways to find the integral (the commutation method and the partial integration method.

The problem in the diagram is to find a bridge to solve a differential equation.

Balance against pickpockets'There is a special formula for the case of +p(x)y q(x):

p(x) =1

e^[∫p(x) dx] =e^(-x)

dy/dx -y = x

The lead guess on both sides is multiplied by e (-x).

e^(-x).[dy/dx -y ] x. e^(-x)d/dx ( x. e^(-x)

Huaiju type x e^(-x) dx

Equip x de (-x).

e^(-x) dx

e^(-x) +c

y=-x -1 +

Composite functions. The situation varies widely, and it is usually reduced to a simple basic function and then integrated. For example, (sinx) 2dx = 1-cos2x) 2]dx = dx 2-(1 2) cos2xdx =x 2-(sin2x 2) 2+c =x 2-sin2x 4+c can be integrated into an infinite series and then the generation will not get a simple elementary function.