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  <div class="t">

         <h5 class="t1">含有<span>ax+b</span>的积分</h5>


      
<table class="table ">
   
    <tr  >
      
        <th  >积分表</th>
        
   
    </tr>

      <tr>
        <td>
     <math>\int\ (ax+b)^n {d}x=\frac{(ax+b)^{n+1}}{a(n+1)}+C</math>
        </td> 
    </tr>



      <tr>
        <td>
     <math>\int\frac{1}{ax+b} {d}x=\frac{1}{a}\ln \left | ax+b\right|+C</math>
        </td> 
    </tr>



    
      <tr>
        <td>
     <math>\int\frac{x}{ax+b} {d}x=\frac{1}{a^2}(ax+b-b\ln\left |ax+b \right|) +C</math>
        </td> 
    </tr>




    
      <tr>
        <td>
     <math>\int\frac{x^2}{ax+b} {d}x=\frac{1}{2a^3} \left[(ax+b)^2-4b(ax+b)+2b^2 \ln\left |ax+b\right| \right]+C</math>
        </td> 
    </tr>



    
      <tr>
        <td>
     <math>\int\frac{1}{x(ax+b)} {d}x = -\frac{1}{b}\ln\left | \frac{ax+b}{x}\right |+C</math>
        </td> 
    </tr>
    
      <tr>
        <td>
     <math>\int\frac{1}{x^2(ax+b)} {d}x=\frac{a}{b^2}\ln\left |\frac{ax+b}{x}\right |-\frac{1}{bx}+C</math>
        </td> 
    </tr>
     

    </table>
   </div>















     <div class="t">

         <h5 class="t1">含有<span>\sqrt{a+bx}</span>的积分</h5>


      
<table class="table ">
   
    <tr  >
      
        <th  >积分表</th>
        
   
    </tr>

      <tr>
        <td>
<math>\int x\sqrt{a+bx} {d}x=\frac{2}{15b^2}(3bx-2a)(a+bx)^{\frac{3}{2}}+C</math>
        </td> 
    </tr>



      <tr>
        <td>
    <math>\int x^2\sqrt{a+bx} {d}x=\frac{2}{105b^3}(15b^2x^2-12abx+8a^2)(a+bx)^{\frac{3}{2}}+C</math>
        </td> 
    </tr>



    
      <tr>
        <td>
 <math>\int x^n\sqrt{a+bx} {d}x=\frac{2}{b(2n+3)}x^n(a+bx)^{\frac{3}{2}} -\frac{2na}{b(2n+3)}\int x^{n-1}\sqrt{a+bx} {d}x</math>
        </td> 
    </tr>




    
      <tr>
        <td>
    <math>\int\frac{\sqrt{a+bx}}{x} {d}x=2\sqrt{a+bx}+a\int\frac{1}{x\sqrt{a+bx}} {d}x</math>
        </td> 
    </tr>



    
      <tr>
        <td>
    <math>\int\frac{\sqrt{a+bx}}{x^n} {d}x=\frac{-1}{a(n-1)}\frac{(a+bx)^{\frac{3}{2}}}{x^{n-1}} -\frac{(2n-5)b}{2a(n-1)}\int\frac{\sqrt{a+bx}}{x^{n-1}} {d}x,n\neq 1</math>
        </td> 
    </tr>
    
      <tr>
        <td>
 <math>\int\frac{1}{x\sqrt{a+bx}} {d}x=\frac{1}{\sqrt{a}}\ln\left (\frac{\sqrt{a+bx} -\sqrt{a}}{\sqrt{a+bx}+\sqrt{a}}\right )+C,a>0</math>
        </td> 
    </tr>
     



      <tr>
        <td>
 <math> \int\frac{1}{x\sqrt{a+bx}} {d}x =\frac{2}{\sqrt{-a}}\arctan\sqrt\frac{a+bx}{-a} +C,a<0</math>
        </td> 
    </tr>
     
    
      <tr>
        <td>
 <math>\int\frac{x}{\sqrt{a+bx}} {d}x=\frac{2(a+bx)^{\frac{3}{2}}}{3b^2} -\frac{(2a)\sqrt{a+bx}}{b^2}</math>
        </td> 
    </tr>


    
      <tr>
        <td>
 <math>\int\frac{1}{x^n\sqrt{a+bx}} {d}x=\frac{-1}{a(n-1)}\frac{\sqrt{a+bx}}{x^{n-1}} -\frac{(2n-3)b}{2a(n-1)}\int\frac{1}{x^{n-1}}\sqrt{a+bx} {d}x,n\neq 1</math>
        </td> 
    </tr>
    </table>
   </div>





    


     <div class="t">

         <h5 class="t1">含有<span>x^2\pm\alpha^2</span>的积分</h5>


      
<table class="table ">
   
    <tr  >
      
        <th  >积分表</th>
        
   
    </tr>

    <tr>
        <td>
            <math>\int\frac{1}{x^2+\alpha^2} {d}x=\frac{\arctan\dfrac{x}{\alpha}}{\alpha}+C</math>
        </td>
    </tr>


      <tr>
        <td>
            <math>\int\frac{1}{\pm x^2\mp\alpha^2} {d}x = \frac{\ln\left(\dfrac{x\mp\alpha}{\pm x+\alpha}\right)}{2\alpha}+C</math>
        </td>
    </tr>


    </table>
         </div>







    
     <div class="t">

         <h5 class="t1"> 含有 <span>{ax^2+b}</span>的积分</h5>


      
<table class="table ">
   
    <tr  >
      
        <th  >积分表</th>
        
   
    </tr>

    <tr>
        <td>
<math>\int\frac{1}{ax^2+b} {d}x=\frac{1}{\sqrt{ab}} \arctan\frac{\sqrt{a}x}{\sqrt{b}}+C</math>
        </td>
    </tr>
    </table>
         </div>
















    
    
     <div class="t">

         <h5 class="t1"> 含有 <span>ax^2+bx+c</span>的积分 </h5>


      
<table class="table ">
   
    <tr  >
      
        <th  >积分表</th>
        
   
    </tr>

    <tr>
        <td>
<math>\int ax^2+bx+c {d}x=\frac{ax^3}{3}+\frac{bx^2}{2}+cx+C</math>
        </td>
    </tr>
    </table>
         </div>

     




















    
    
     <div class="t">

         <h5 class="t1"> 含有 <math>\sqrt{a^2+x^2}\qquad(a>0)</math>的积分 </h5>


      
<table class="table ">
   
    <tr  >
      
        <th  >积分表</th>
        
   
    </tr>

    <tr>
        <td>
            <math>\int\sqrt{a^2+x^2} {d}x=\frac{1}{2}x\sqrt{a^2+x^2}+\frac{1}{2}a^2\ln\left (x+\sqrt{a^2+x^2}\right )+C</math>
 
        </td>
    </tr>



      <tr>
        <td>
 <math>\int x^2\sqrt{a^2+x^2} {d}x=\frac{1}{8}x(a^2+2x^2)\sqrt{a^2+x^2}-\frac{1}{8}a^4\ln\left (x+\sqrt{a^2+x^2}\right )+C</math>
        </td>
    </tr>





      <tr>
        <td>
 <math>\int \frac{\sqrt{a^2+x^2}}{x} {d}x = \sqrt{a^2+x^2} - a\ln \left ( \frac{a+\sqrt{a^2+x^2}}{x} \right ) +C</math>
        </td>
    </tr>





      <tr>
        <td>
 <math>\int \frac{\sqrt{a^2+x^2}}{x^2} {d}x = \ln\left ( x+\sqrt{a^2+x^2}\right ) -  \frac{\sqrt{a^2+x^2}}{x} +C</math>
        </td>
    </tr>





      <tr>
        <td>
 <math>\int \frac{1}{\sqrt{a^2+x^2}} {d}x=\ln \left ( x+\sqrt{a^2+x^2} \right ) +C</math>
        </td>
    </tr>



      <tr>
        <td>
 <math>\int\frac{x^2}{\sqrt{a^2+x^2}} {d}x=\frac{1}{2}x\sqrt{a^2+x^2} -\frac{1}{2}a^2\ln\left (\sqrt{a^2+x^2}+x \right )+C</math>
        </td>
    </tr>




      <tr>
        <td>
 <math>\int\frac{1}{x\sqrt{a^2+x^2}} {d}x=\frac{1}{a}\ln\left (\frac{x}{a+\sqrt{a^2+x^2}}\right )+C</math>
        </td>
    </tr>




      <tr>
        <td>
 <math>\int\frac{1}{x^2\sqrt{a^2+x^2}} {d}x=-\frac{\sqrt{a^2+x^2}}{a^2x}+C</math>
        </td>
    </tr>
      
    </table>
         </div>

    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
    
     <div class="t">

         <h5 class="t1"> 含有<span>\sqrt{x^2-a^2}\qquad{(x^2>a^2)}</span>的积分 </h5>


      
<table class="table ">
   
    <tr  >
      
        <th  >积分表</th>
        
   
    </tr>

    <tr>
        <td>
          <math>\int \frac{1}{\sqrt{x^2-a^2}} {d}x=\ln| x+\sqrt{x^2-a^2} | +C</math>
 
        </td>
    </tr>

    </table>
         </div>
    
    
    
    
    
    
    
    
    
    
    
    




















    
    
    
     <div class="t">

         <h5 class="t1"> 含有<span>\sqrt{a^2-x^2} \qquad(a^2>x^2)</span>的积分 </h5>


      
<table class="table ">
   
    <tr  >
      
        <th  >积分表</th>
        
   
    </tr>

    <tr>
        <td>
          
 <math>\int\sqrt{a^2-x^2} {d}x=\frac{1}{2}x\sqrt{a^2-x^2} +\frac{a^2}{2}\arcsin\frac{x}{a}+C</math>
        </td>
    </tr>


      <tr>
        <td>
          <math>\int \frac{1}{\sqrt{a^2-x^2}} {d}x= \arcsin \frac{x}{a} +C = - \arccos \frac{x}{a} +C</math>
 
        </td>
    </tr>


      <tr>
        <td>
          <math>\int x^2\sqrt{a^2-x^2} {d}x=\frac{1}{8}x(2x^2-a^2)\sqrt{a^2-x^2} +\frac{1}{8}a^4\arcsin\frac{x}{a}+C</math>
 
        </td>
    </tr>


      <tr>
        <td>
          <math>\int\frac{\sqrt{a^2-x^2}}{x} {d}x=\sqrt{a^2-x^2} -a\ln\left (\frac{a+\sqrt{a^2-x^2}}{x}\right )+C</math>
 
        </td>
    </tr>


      <tr>
        <td>
          <math>\int\frac{\sqrt{a^2-x^2}}{x^2} {d}x=-\frac{\sqrt{a^2-x^2}}{x} -\arcsin\frac{x}{a}+C</math>
 
        </td>
    </tr>


      <tr>
        <td>
          <math>\int\frac{1}{x\sqrt{a^2-x^2}} {d}x=-\frac{1}{a}\ln\left (\frac{a+\sqrt{a^2-x^2}}{x}\right )+C</math>
 
        </td>
    </tr>


      <tr>
        <td>
          <math>\int\frac{x^2}{\sqrt{a^2-x^2}} {d}x=-\frac{1}{2}x\sqrt{a^2-x^2}+\frac{1}{2}a^2\arcsin\frac{x}{a}+C</math>
 
        </td>
    </tr>


      <tr>
        <td>
          <math>\int\frac{1}{x^2\sqrt{a^2-x^2}} {d}x=-\frac{\sqrt{a^2-x^2}}{a^2x}+C</math>
 
        </td>
    </tr>


    </table>
         </div>
    
    
    
    
    
    
    
    
   <!-- 12-->
    

        
     <div class="t">

         <h5 class="t1"> 含有<span>R=\sqrt{|a|x^2+bx+c}\qquad(a\ne0)</span>的积分 </h5>


      
<table class="table ">
   
    <tr  >
      
        <th  >积分表</th>
        
   
    </tr>

    <tr>
        <td>
            <math>\int\frac{ {d}x}{R} = \frac{1}{\sqrt{a}}\ln\left(2\sqrt{a}R+2ax+b\right)\qquad( {for }a>0)</math>
 
        </td>
    </tr>


      <tr>
        <td>
       <math>\int\frac{ {d}x}{R} = \frac{1}{\sqrt{a}}\,\operatorname{arsinh}\frac{2ax+b}{\sqrt{4ac-b^2}} \qquad  {(for }a>0 {, }4ac-b^2>0 {)}</math>
 
        </td>
    </tr>


      <tr>
        <td>
         <math>\int\frac{ {d}x}{R} = \frac{1}{\sqrt{a}}\ln|2ax+b| \quad  {(for }a>0 {, }4ac-b^2=0 {)}</math>
 
        </td>
    </tr>


      <tr>
        <td>
           <math>\int\frac{ {d}x}{R} = -\frac{1}{\sqrt{-a}}\arcsin\frac{2ax+b}{\sqrt{b^2-4ac}} \qquad  {(for }a<0 {, }4ac-b^2<0 {, }\left(2ax+b\right)<\sqrt{b^2-4ac} {)}</math>
 
        </td>
    </tr>


      <tr>
        <td>
            <math>\int\frac{ {d}x}{R^3} = \frac{4ax+2b}{(4ac-b^2)R}</math>
 
        </td>
    </tr>


      <tr>
        <td>
 <math>\int\frac{ {d}x}{R^5} = \frac{4ax+2b}{3(4ac-b^2)R}\left(\frac{1}{R^2}+\frac{8a}{4ac-b^2}\right)</math>
 
        </td>
    </tr>


      <tr>
        <td>
         <math>\int\frac{ {d}x}{R^{2n+1}} = \frac{2}{(2n-1)(4ac-b^2)}\left[\frac{2ax+b}{R^{2n-1}}+4a(n-1)\int\frac{ {d}x}{R^{2n-1}}\right]</math>
 
        </td>
    </tr>


      <tr>
        <td>
        <math>\int\frac{x}{R}\; {d}x = \frac{R}{a}-\frac{b}{2a}\int\frac{ {d}x}{R}</math>
 
        </td>
    </tr>



    
      <tr>
        <td>
        
 <math>\int\frac{x}{R^3}\; {d}x = -\frac{2bx+4c}{(4ac-b^2)R}</math>
        </td>
    </tr>




    
      <tr>
        <td>
        <math>\int\frac{x}{R^{2n+1}}\; {d}x = -\frac{1}{(2n-1)aR^{2n-1}}-\frac{b}{2a}\int\frac{ {d}x}{R^{2n+1}}</math>
 
        </td>
    </tr>



    
      <tr>
        <td>
        <math>\int\frac{ {d}x}{xR}=-\frac{1}{\sqrt{c}}\ln\left(\frac{2\sqrt{c}R+bx+2c}{x}\right)</math>
 
        </td>
    </tr>



    
      <tr>
        <td>
        <math>\int\frac{ {d}x}{xR}=-\frac{1}{\sqrt{c}}\operatorname{arsinh}\left(\frac{bx+2c}{|x|\sqrt{4ac-b^2}}\right)</math>
 
        </td>
    </tr>


    </table>
         </div>
    
    


       <!-- 三角-->




    
        
     <div class="t">

         <h5 class="t1">含有三角函数的积分 </h5>


      
<table class="table ">
   
    <tr  >
      
        <th  >积分表</th>
        
   
    </tr>




<tr><td><math>\int\cos x {d}x=\sin x+C</math></td></tr>
<tr><td><math>\int\sin x {d}x= -\cos x+C</math></td></tr>
<tr><td><math>\int\sec^2x {d}x=\tan x+C</math></td></tr>
<tr><td><math>\int\csc^2x {d}x=-\cot x+C</math></td></tr>
<tr><td><math>\int\sec x\tan x {d}x=\sec x+C</math></td></tr>
<tr><td><math>\int\csc x\cot x {d}x=-\csc x+C</math></td></tr>

<tr><td><math>\int\tan x {d}x=-\ln{\left|\cos {x}\right|}+C=\ln{\left|\sec x\right|}+C</math></td></tr>
<tr><td><math>\int\cot x {d}x=\ln{\left|\sin x\right|}+C</math></td></tr>
<tr><td><math>\int\sec x {d}x=\ln{\left|\sec x+\tan x\right|}+C</math></td></tr>
<tr><td><math>\int\csc x {d}x=\ln{\left|\csc x-\cot x\right|}+C=\ln{\left|{\tan x-\sin x\over\sin x\tan x}\right|}+C</math></td></tr>


<tr><td><math>\int \sin ^n x  {d}x = - \frac{1}{n} \sin ^{n-1} x \cos x + \frac{n-1}{n} \int \sin ^{n-2} x  {d}x +C \quad \forall n \ge 2</math></td></tr>
<tr><td><math>\int \sin ^2 x  {d}x = \frac{x}{2}-\frac{\sin{2x}}{4} +C</math></td></tr>


<tr><td><math>\int \cos ^n x  {d}x = \frac{1}{n} \cos ^{n-1} x \sin x + \frac{n-1}{n} \int \cos ^{n-2} x  {d}x +C \quad \forall n \ge 2</math></td></tr>
<tr><td><math>\int \cos ^2 x  {d}x = \frac{x}{2}+\frac{\sin{2x}}{4} +C</math></td></tr>


<tr><td><math>\int \tan ^n x  {d}x = \frac{1}{n-1} \tan ^{n-1} x - \int \tan ^{n-2} x  {d}x +C \quad \forall n \ge 2</math></td></tr>
<tr><td><math>\int \tan ^2 x  {d}x = \tan x - x +C</math></td></tr>


<tr><td><math>\int \cot ^n x  {d}x = - \frac{1}{n-1} \cot ^{n-1} x - \int \cot ^{n-2} x  {d}x +C \quad \forall n \ge 2</math></td></tr>
<tr><td><math>\int \cot ^2 x  {d}x = - \cot x - x +C</math></td></tr>


<tr><td><math>\int \sec ^n x  {d}x = \frac{1}{n-1} \sec ^{n-2} x \tan x + \frac{n-2}{n-1} \int \sec ^{n-2} x  {d}x +C \quad \forall n \ge 2</math></td></tr>


<tr><td><math>\int \csc ^n x  {d}x = - \frac{1}{n-1} \csc ^{n-2} x \cot x + \frac{n-2}{n-1} \int \csc ^{n-2} x  {d}x +C \quad \forall n \ge 2</math></td></tr>

 



 

    </table>
         </div>
    
    



    

    <!-- 含有反三角函数的积分 -->

        
     <div class="t">

         <h5 class="t1">含有反三角函数的积分 </h5>

    
<table class="table ">
   
    <tr  >
      
        <th  >积分表</th>
        
   
    </tr> 

    <tr><td><math>\int \arcsin x  {d}x = x \arcsin x + \sqrt {1 - x^2} +C</math></td></tr>
<tr><td><math>\int \arccos x  {d}x = x \arccos x - \sqrt {1 - x^2} +C</math></td></tr>
<tr><td><math>\int \arctan{x} \, dx = x \arctan{x} - \frac{1}{2} \ln { \vert 1 + x^2 \vert } + C</math></td></tr>
<tr><td><math>\int arccot{x} \, dx = x arccot{x} + \frac{1}{2} \ln { \vert 1 + x^2 \vert } + C</math></td></tr>
<tr><td><math>\int arcsec x  {d}x = x arcsec x - sgn(x)\ln \left|x + \sqrt{x^2 - 1}\right| +C
                                = x arcsec x + sgn(x)\ln \left|x - \sqrt{x^2 - 1}\right| +C</math></td></tr>
<tr><td><math>\int arccsc x  {d}x = x arccsc x + sgn(x)\ln \left|x + \sqrt{x^2 - 1}\right| +C
                                = x arccsc x - sgn(x)\ln \left|x - \sqrt{x^2 - 1}\right| +C</math></td></tr>


    </table>
    </div>







    <!-- 含有指数函数的积分 -->
    
     <div class="t">

         <h5 class="t1">含有指数函数的积分 </h5>

    
<table class="table ">
   
    <tr  > 
        <th  >积分表</th> 
    </tr> 

<tr><td><math>\int e^x {d}x=e^x+C</math></td></tr>
<tr><td><math>\int\alpha^x {d}x=\frac{\alpha^x}{\ln\alpha}+C</math></td></tr>
<tr><td><math>\int xe^{ax} {d}x=\frac{1}{a^2}(ax-1)e^{ax}+C</math></td></tr>
<tr><td><math>\int x^ne^{ax} {d}x=\frac{1}{a}x^ne^{ax}-\frac{n}{a}\int x^{n-1}e^{ax} {d}x</math></td></tr>
<tr><td><math>\int e^{ax}\sin bx  {d}x=\frac{e^{ax}}{a^2+b^2}(a\sin bx-b\cos bx)+C</math></td></tr>
<tr><td><math>\int e^{ax}\cos bx  {d}x=\frac{e^{ax}}{a^2+b^2}(a\cos bx+b\sin bx)+C</math></td></tr>
 
     
    </table>
    </div>

















         <div class="t">

         <h5 class="t1"> 含有对数函数的积分  </h5>

    
<table class="table ">
   
    <tr  > 
        <th  >积分表</th> 
    </tr> 

    <tr>
        <td>
<math>\int\ln x {d}x = x\ln x - x + C</math>

        </td>
    </tr>



       <tr>
        <td>
<math>\int\log_\alpha x {d}x=\frac{1}{\ln\alpha}\left({x\ln x - x}\right)+C</math>

        </td>
    </tr>



       <tr>
        <td>

<math>\int x^n\ln x {d}x = \frac{x^{n+1}}{(n+1)^2}[(n+1)\ln x -1]+ C</math>
        </td>
    </tr>


       <tr>
        <td>

<math>\int\frac{1}{x\ln{x}} {d}x = \ln{(\ln{x})}+C</math>
        </td>
    </tr>
    </table>
 </div>












    
         <div class="t">

         <h5 class="t1"> 含有双曲函数的积分  </h5>

    
<table class="table ">
   
    <tr  > 
        <th  >积分表</th> 
    </tr> 

    <tr>
        <td>
 <math>\int \sinh x  {d}x = \cosh x +C</math>

        </td>
    </tr>



       <tr>
        <td>
<math>\int \cosh x  {d}x = \sinh x +C</math>

        </td>
    </tr>



       <tr>
        <td>

<math>\int \tanh x  {d}x = \ln\left(\cosh x\right) +C</math>
        </td>
    </tr>


       <tr>
        <td>

<math>\int \coth x  {d}x = \ln\left|\sinh x\right| +C</math>
        </td>
    </tr>





          <tr>
        <td>
<math>\int  {sech}\ x  {d}x = \arcsin\left(\tanh x\right) + C =  \arctan\left(\sinh x\right) + C </math>
 
        </td>
    </tr>

          <tr>
        <td>
<math>\int  {csch}\ x  {d}x = \ln\left|\tanh {x \over2}\right| + C</math>
 
        </td>
    </tr>
    </table>
 </div>


























           <div class="t">

         <h5 class="t1"> 定积分  </h5>

    
<table class="table ">
   
    <tr  > 
        <th  >积分表</th> 
    </tr> 

    <tr>
        <td>
<math>\int^\infty_{-\infty}e^{-\alpha x^2} {d}x=\sqrt{\frac{\pi}{\alpha}}</math>

        </td>
    </tr>



       <tr>
        <td>
<math>\int_0^\frac{\pi}{2}  {sin}^n x {d}x=\int_0^\frac{\pi}{2}  {cos}^n x {d}x=
\begin{cases}
\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdot\ldots\cdot\frac{4}{5}\cdot\frac{2}{3}, &  {if }n>1 { 且 }n { 为奇数  } \\
\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdot\ldots\cdot\frac{3}{4}\cdot\frac{1}{2}\cdot\frac{\pi}{2}, &  {if }n>0 { 且 }n { 为偶数  }
\end{cases}

</math>

        </td>
    </tr>


     
    </table>
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