Univariate function integration_ Partial integral method

Think about a problem  

analysis ： You can use the derivation rule of the product of two functions , 

Let's set the function u(x) and v(x) Have continuous derivatives ,  (uv)' = u'v  +  uv' 

Deform the above formula by ：uv' = (uv)' - u'v.

Take integral on both sides , have to ： 

∫ uv' dx = uv —  ∫ u'vdx .    This equation is the partial integral formula .

One .  Definition of partial integral formula

Now? , We can figure out , To calculate an integral , It can be expressed as seeking uv Subtract another integral .

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Be careful ： v'dx = dv,   u'dx = du.    That is to say d Then follow the original function , Originally there were v',  It's converted into dv, 

Originally there were u', It's converted into du.

So the partial integral formula Can also be written as ：

∫ udv = uv - ∫ vdu

Now compare :

Two .  With the partial integral formula ,  Now we need to consider which function is u,     Which function is used as v？

Let's look at an example

From the above two methods , It can be seen that , Set up u, v The requirements of

（1） v It should be easy to find

（2）∫vdu   than  ∫udv Easy to seek . 

3、 ... and . Now back to the topic mentioned at the beginning of this article

Solve with partial integral method

Explain ： Make u=x,   v' = eˣ, 

  ∫xeˣdx = xeˣ -  ∫eˣdx = xeˣ - eˣ + C

Four .  Conclusion of integral with power function

1. If it is a power function multiplied by a trigonometric function , choose u Is a power function , v' It's a trigonometric function  

2.  If it is a power function multiplied by an exponential function , choose u Is a power function , v' It's an exponential function  

Set it like this , Is to reduce the power by one .

5、 ... and . Integration of trigonometric function and exponential function

  Encounter the integral of trigonometric function multiplied by exponential function ,  set up u For any of these functions .

6、 ... and . Integral of inverse trigonometric function

For anti triangle , There is such a pithy formula ： The easy integral is set to v ,  The easy derivative is set to u.

Answer this question ,    Remember  (arctanx)'= 1/ (1+x²) 

Conclusion ：  In the integral of inverse trigonometric function ,  selected u Is an inverse trigonometric function , Because the original function of the inverse triangle is difficult to find .

7、 ... and .  Integral of logarithmic function

Conclusion ：  When there is a logarithmic function in the integrand ,  choose u It's a logarithmic function .

example ： Find integral  ∫x³ lnxdx.

Explain ： If you will v' Set to lnx,  Can we solve the original function v Well ？ It's hard ！

    So we're going to u Set to lnx,   v' = x³

    simple form

8、 ... and .    Sum up the integrals of the above functions ,

choose u The priority of is ： Against power three   perhaps   Anti power three finger