Integrals

Single Integrals

Space Curves:

Arclength:  $\underset{a}{\overset{b}{\int }}\Vert \overrightarrow{r}(t)\Vert dt$

Scalar Line Integral:  $\underset{C}{\int }fds=\underset{a}{\overset{b}{\int }}f(\overrightarrow{r}(t))*\Vert \overrightarrow{r}(t{)}^{\prime }\Vert dt$

Integral Forms & Antiderivatives

Basic Forms:

1. $\int udv=uv-\int vdu$
   
2. $\int u^{n}du=\frac{u^{n+1}}{n+1}+C,(n\ne -1)$
   
3. $\int \frac{du}{u}=\ln |u|+C$
   
4. $\int e^{u}du=e^u+C$
   
5. $\int a^{u}du=\frac{a^u}{\ln a}+C$
   
6. $\int \sin udu=-\cos u+C$
   
7. $\int \cos udu=\sin u+C$
   
8. $\int {\sec }^{2}udu=\tan u+C$
   
9. $\int {\csc }^{2}udu=-\cot u+C$
   
10. $\int \sec u\tan udu=\sec u+C$
    
11. $\int \csc u\cot udu=-\csc u+C$
    
12. $\int \tan udu=\ln |\sec u|+C$
    
13. $\int \cot udu=\ln |\sin u|+C$
    
14. $\int \sec udu=\ln |\sec u+\tan u|+C$
    
15. $\int \csc udu=\ln |\csc u-\cot u|+C$
    
16. $\int \frac{du}{\sqrt{a^2-u^2}}={\sin }^{-1}\frac{u}{a}+C,(a>0)$
    
17. $\int \frac{du}{a^2+u^2}=\frac{1}{a}{\tan }^{-1}\frac{u}{a}+C$
    
18. $\int \frac{du}{u\sqrt{u^2-a^2}}=\frac{1}{a}{\sec }^{-1}\frac{u}{a}+C$
    
19. $\int \frac{du}{a^2-u^2}=\frac{1}{2a}\ln \left|\frac{u+a}{u-a}\right|+C$
    
20. $\int \frac{du}{u^2-a^2}=\frac{1}{2a}\ln \left|\frac{u-a}{u+a}\right|+C$
    

Forms Involving: $\sqrt{a^2+u^2},a>0$

1. $\int \sqrt{a^2+u^2}du=\frac{u}{2}\sqrt{a^2+u^2}+\frac{a^2}{2}\ln (u+\sqrt{a^2+u^2})+C$
   
2. $\int u^2\sqrt{a^2+u^2}du=\frac{u}{8}(a^2+2u^2)\sqrt{a^2+u^2}-\frac{a^4}{8}\ln (u+\sqrt{a^2+u^2})+C$
   
3. $\int \frac{\sqrt{a^2+u^2}}{u}du=\sqrt{a^2+u^2}-a\ln \left|\frac{a+\sqrt{a^2+u^2}}{u}\right|+C$
   
4. $\int \frac{\sqrt{a^2+u^2}}{u^2}du=-\frac{\sqrt{a^2+u^2}}{u}+\ln (u+\sqrt{a^2+u^2})+C$
   
5. $\int \frac{du}{\sqrt{a^2+u^2}}=\ln (u+\sqrt{a^2+u^2})+C$
   
6. $\int \frac{u^{2}du}{\sqrt{a^2+u^2}}=\frac{u}{2}\sqrt{a^2+u^2}-\frac{a^2}{2}\ln (u+\sqrt{a^2+u^2})+C$
   
7. $\int \frac{du}{u\sqrt{a^2+u^2}}=-\frac{1}{a}\ln \left|\frac{\sqrt{a^2+u^2}+a}{u}\right|+C$
   
8. $\int \frac{du}{u^2\sqrt{a^2+u^2}}=-\frac{\sqrt{a^2+u^2}}{a^{2}u}+C$
   
9. $\int \frac{du}{(a^2+u^2{)}^{3/2}}=\frac{u}{a^2\sqrt{a^2+u^2}}+C$
   

Forms Involving: $\sqrt{a^2-u^2},a>0$

1. $\int \sqrt{a^2-u^2}du=\frac{u}{2}\sqrt{a^2-u^2}+\frac{a^2}{2}{\sin }^{-1}\frac{u}{a}+C$
   
2. $\int u^2\sqrt{a^2-u^2}du=\frac{u}{8}(2u^2-a^2)\sqrt{a^2-u^2}+\frac{a^4}{8}{\sin }^{-1}\frac{u}{a}+C$
   
3. $\int \frac{\sqrt{a^2-u^2}}{u}du=\sqrt{a^2-u^2}-a\ln \left|\frac{a+\sqrt{a^2-u^2}}{u}\right|+C$
   
4. $\int \frac{\sqrt{a^2-u^2}}{u^2}du=-\frac{1}{u}\sqrt{a^2-u^2}-{\sin }^{-1}\frac{u}{a}+C$
   
5. $\int \frac{u^{2}du}{\sqrt{a^2-u^2}}=-\frac{u}{2}\sqrt{a^2-u^2}+\frac{a^2}{2}{\sin }^{-1}\frac{u}{a}+C$
   
6. $\int \frac{du}{u\sqrt{a^2-u^2}}=-\frac{1}{a}\ln \left|\frac{a+\sqrt{a^2-u^2}}{u}\right|+C$
   
7. $\int \frac{du}{u^2\sqrt{a^2-u^2}}=-\frac{1}{a^{2}u}\sqrt{a^2-u^2}+C$
   
8. $\int (a^2-u^2{)}^{3/2}du=-\frac{u}{8}(2u^2-5a^2)\sqrt{a^2-u^2}+\frac{3a^4}{8}{\sin }^{-1}\frac{u}{a}+C$
   
9. $\int \frac{du}{(a^2-u^2{)}^{3/2}}=\frac{u}{a^2\sqrt{a^2-u^2}}+C$
   

Forms Involving: $\sqrt{u^2-a^2},a>0$

1. $\int \sqrt{u^2-a^2}du=\frac{u}{2}\sqrt{u^2-a^2}-\frac{a^2}{2}\ln |u+\sqrt{u^2-a^2}|+C$
   
2. $\int u^2\sqrt{u^2-a^2}du=\frac{u}{8}(2u^2-a^2)\sqrt{u^2-a^2}-\frac{a^4}{8}\ln |u+\sqrt{u^2-a^2}|+C$
   
3. $\int \frac{\sqrt{u^2-a^2}}{u}du=\sqrt{u^2-a^2}-a{\cos }^{-1}\frac{a}{|u|}+C$
   
4. $\int \frac{\sqrt{u^2-a^2}}{u^2}du=-\frac{\sqrt{u^2-a^2}}{u}+\ln |u+\sqrt{u^2-a^2}|+C$
   
5. $\int \frac{du}{\sqrt{u^2-a^2}}=\ln |u+\sqrt{u^2-a^2}|+C$
   
6. $\int \frac{u^{2}du}{\sqrt{u^2-a^2}}=\frac{u}{2}\sqrt{u^2-a^2}+\frac{a^2}{2}\ln |u+\sqrt{u^2-a^2}|+C$
   
7. $\int \frac{du}{u^2\sqrt{u^2-a^2}}=\frac{\sqrt{u^2-a^2}}{a^{2}u}+C$
   
8. $\int \frac{du}{(u^2-a^2{)}^{3/2}}=-\frac{u}{a^2\sqrt{u^2-a^2}}+C$
   

Forms Involving: $a+bu$

1. $\int \frac{udu}{a+bu}=\frac{1}{b^2}(a+bu-a\ln |a+bu|)+C$
2. $\int \frac{u^{2}du}{a+bu}=\frac{1}{2b^3}[(a+bu{)}^2-4a(a+bu)+2a^2\ln |a+bu|]+C$
3. $\int \frac{du}{u(a+bu)}=\frac{1}{a}\ln \left|\frac{u}{a+bu}\right|+C$
4. $\int \frac{du}{u^2(a+bu)}=-\frac{1}{au}+\frac{b}{a^2}\ln \left|\frac{a+bu}{u}\right|+C$
5. $\int \frac{udu}{(a+bu{)}^2}=\frac{a}{b^2(a+bu)}+\frac{1}{b^2}\ln |a+bu|+C$
6. $\int \frac{du}{u(a+bu{)}^2}=\frac{1}{a(a+bu)}-\frac{1}{a^2}\ln \left|\frac{a+bu}{u}\right|+C$
7. $\int \frac{u^{2}du}{(a+bu{)}^2}=\frac{1}{b^3}(a+bu-\frac{a^2}{a+bu}-2a\ln |a+bu|)+C$
8. $\int u\sqrt{a+bu}du=\frac{2}{15b^2}(3bu-2a)(a+bu{)}^{3/2}+C$
9. $\int \frac{udu}{\sqrt{a+bu}}=\frac{2}{3b^2}(bu-2a)\sqrt{a+bu}+C$
10. $\int \frac{u^{2}du}{\sqrt{a+bu}}=\frac{2}{15b^3}(8a^2+3b^2{u}^2-4abu)\sqrt{a+bu}+C$
11. $\int \frac{du}{u\sqrt{a+bu}}=\{\begin{array}{cc}\frac{1}{\sqrt{a}}\ln \left|\frac{\sqrt{a+bu}-\sqrt{a}}{\sqrt{a+bu}+\sqrt{a}}\right|+C& (\text{if }a>0)\\ \frac{2}{\sqrt{-a}}{\tan }^{-1}\sqrt{\frac{a+bu}{-a}}+C& (\text{if }a<0)\end{array}$
12. $\int \frac{\sqrt{a+bu}}{u}du=2\sqrt{a+bu}+a\int \frac{du}{u\sqrt{a+bu}}$
13. $\int \frac{\sqrt{a+bu}}{u^2}du=-\frac{\sqrt{a+bu}}{u}+\frac{b}{2}\int \frac{du}{u\sqrt{a+bu}}$
14. $\int u^n\sqrt{a+bu}du=\frac{2}{b(2n+3)}[u^n(a+bu{)}^{3/2}-na\int u^{n-1}\sqrt{a+bu}du]$
15. $\int \frac{u^{n}du}{\sqrt{a+bu}}=\frac{2u^n\sqrt{a+bu}}{b(2n+1)}-\frac{2na}{b(2n+1)}\int \frac{u^{n-1}du}{\sqrt{a+bu}}$
16. $\int \frac{du}{u^n\sqrt{a+bu}}=-\frac{\sqrt{a+bu}}{a(n-1)u^{n-1}}-\frac{b(2n-3)}{2a(n-1)}\int \frac{du}{u^{n-1}\sqrt{a+bu}}$

Trigonometric Forms:

1. $\int {\sin }^{2}udu=\frac{1}{2}u-\frac{1}{4}\sin 2u+C$
2. $\int {\cos }^{2}udu=\frac{1}{2}u+\frac{1}{4}\sin 2u+C$
3. $\int {\tan }^{2}udu=\tan u-u+C$
4. $\int {\cot }^{2}udu=-\cot u-u+C$
5. $\int {\sin }^{3}udu=-\frac{1}{3}(2+{\sin }^{2}u)\cos u+C$
6. $\int {\cos }^{3}udu=\frac{1}{3}(2+{\cos }^{2}u)\sin u+C$
7. $\int {\tan }^{3}udu=\frac{1}{2}{\tan }^{2}u+\ln |\cos u|+C$
8. $\int {\cot }^{3}udu=-\frac{1}{2}{\cot }^{2}u-\ln |\sin u|+C$
9. $\int {\sec }^{3}udu=\frac{1}{2}\sec u\tan u+\frac{1}{2}\ln |\sec u+\tan u|+C$
10. $\int {\csc }^{3}udu=-\frac{1}{2}\csc u\cot u+\frac{1}{2}\ln |\csc u-\cot u|+C$
11. $\int {\sin }^{n}udu=-\frac{1}{n}{\sin }^{n-1}u\cos u+\frac{n-1}{n}\int {\sin }^{n-2}udu$
12. $\int {\cos }^{n}udu=\frac{1}{n}{\cos }^{n-1}u\sin u+\frac{n-1}{n}\int {\cos }^{n-2}udu$
13. $\int {\tan }^{n}udu=\frac{1}{n-1}{\tan }^{n-1}u-\int {\tan }^{n-2}udu$
14. $\int {\cot }^{n}udu=\frac{-1}{n-1}{\cot }^{n-1}u-\int {\cot }^{n-2}udu$
15. $\int {\sec }^{n}udu=\frac{1}{n-1}\tan u{\sec }^{n-2}u+\frac{n-2}{n-1}\int {\sec }^{n-2}udu$
16. $\int {\csc }^{n}udu=\frac{-1}{n-1}\cot u{\csc }^{n-2}u+\frac{n-2}{n-1}\int {\csc }^{n-2}udu$
17. $\int \sin au\sin budu=\frac{\sin (a-b)u}{2(a-b)}-\frac{\sin (a+b)u}{2(a+b)}+C$
18. $\int \cos au\cos budu=\frac{\sin (a-b)u}{2(a-b)}+\frac{\sin (a+b)u}{2(a+b)}+C$
19. $\int \sin au\cos budu=-\frac{\cos (a-b)u}{2(a-b)}-\frac{\cos (a+b)u}{2(a+b)}+C$
20. $\int u\sin udu=\sin u-u\cos u+C$
21. $\int u\cos udu=\cos u+u\sin u+C$
22. $\int u^n\sin udu=-u^n\cos u+n\int u^{n-1}\cos udu$
23. $\int u^n\cos udu=u^n\sin u-n\int u^{n-1}\sin udu$
24. $\int {\sin }^{n}u{\cos }^{m}udu=-\frac{{\sin }^{n-1}u{\cos }^{m+1}u}{n+m}+\frac{n-1}{n+m}\int {\sin }^{n-2}u{\cos }^{m}udu$

 $=\frac{{\sin }^{n+1}u{\cos }^{m-1}u}{n+m}+\frac{m-1}{n+m}\int {\sin }^{n}u{\cos }^{m-2}udu$

Inverse Trigonometric Forms:

1. $\int {\sin }^{-1}udu=u{\sin }^{-1}u+\sqrt{1-u^2}+C$
2. $\int {\cos }^{-1}udu=u{\cos }^{-1}u-\sqrt{1-u^2}+C$
3. $\int {\tan }^{-1}udu=u{\tan }^{-1}u-\frac{1}{2}\ln (1+u^2)+C$
4. $\int u{\sin }^{-1}udu=\frac{2u^2-1}{4}{\sin }^{-1}u+\frac{u\sqrt{1-u^2}}{4}+C$
5. $\int u{\cos }^{-1}udu=\frac{2u^2-1}{4}{\cos }^{-1}u-\frac{u\sqrt{1-u^2}}{4}+C$
6. $\int u{\tan }^{-1}udu=\frac{u^2+1}{2}{\tan }^{-1}u-\frac{u}{2}+C$
7. $\int u^n{\sin }^{-1}udu=\frac{1}{n+1}[u^{n+1}{\sin }^{-1}u-\int \frac{u^{n+1}du}{\sqrt{1-u^2}}],(n\ne -1)$
8. $\int u^n{\cos }^{-1}udu=\frac{1}{n+1}[u^{n+1}{\cos }^{-1}u+\int \frac{u^{n+1}du}{\sqrt{1-u^2}}],(n\ne -1)$
9. $\int u^n{\tan }^{-1}udu=\frac{1}{n+1}[u^{n+1}{\tan }^{-1}u-\int \frac{u^{n+1}du}{1+u^2}],(n\ne -1)$

Exponential and Logarithmic Forms:

1. $\int u{e}^{au}du=\frac{1}{a^2}(au-1)e^{au}+C$
2. $\int u^n{e}^{au}du=\frac{1}{a}{u}^n{e}^{au}-\frac{n}{a}\int u^{n-1}{e}^{au}du$
3. $\int e^{au}\sin budu=\frac{e^{au}}{a^2+b^2}(a\sin bu-b\cos bu)+C$
4. $\int e^{au}\cos budu=\frac{e^{au}}{a^2+b^2}(a\cos bu+b\sin bu)+C$
5. $\int \ln udu=u\ln u-u+C$
6. $\int u^n\ln udu=\frac{u^{n+1}}{(n+1{)}^2}[(n+1)\ln u-1]+C$
7. $\int \frac{1}{u\ln u}du=\ln |\ln u|+C$

Hyperbolic Forms:

1. $\int \sinh udu=\cosh u+C$
2. $\int \cosh udu=\sinh u+C$
3. $\int \tanh udu=\ln \cosh u+C$
4. $\int \coth udu=\ln |\sinh u|+C$
5. $\int \text{sech}udu={\tan }^{-1}|\sinh u|+C$
6. $\int \text{csch}udu=\ln |\tanh \frac{1}{2}u|+C$
7. $\int {\text{sech}}^{2}udu=\tanh u+C$
8. $\int {\text{csch}}^{2}udu=-\coth u+C$
9. $\int \text{sech}u\tanh udu=-\text{sech}u+C$
10. $\int \text{csch}u\coth udu=-\text{csch}u+C$

Forms Involving: $\sqrt{2au-u^2},a>0$

1. $\int \sqrt{2au-u^2}du=\frac{u-a}{2}\sqrt{2au-u^2}+\frac{a^2}{2}{\cos }^{-1}\left(\frac{a-u}{a}\right)+C$
2. $\int u\sqrt{2au-u^2}du=\frac{2u^2-au-3a^2}{6}\sqrt{2au-u^2}+\frac{a^3}{2}{\cos }^{-1}\left(\frac{a-u}{a}\right)+C$
3. $\int \frac{\sqrt{2au-u^2}}{u}du=\sqrt{2au-u^2}+a{\cos }^{-1}\left(\frac{a-u}{a}\right)+C$
4. $\int \frac{\sqrt{2au-u^2}}{u^2}du=-\frac{2\sqrt{2au-u^2}}{u}-{\cos }^{-1}\left(\frac{a-u}{a}\right)+C$
5. $\int \frac{du}{\sqrt{2au-u^2}}={\cos }^{-1}\left(\frac{a-u}{a}\right)+C$
6. $\int \frac{udu}{\sqrt{2au-u^2}}=-\sqrt{2au-u^2}+a{\cos }^{-1}\left(\frac{a-u}{a}\right)+C$
7. $\int \frac{u^{2}du}{\sqrt{2au-u^2}}=-\frac{(u+3a)}{2}\sqrt{2au-u^2}+\frac{3a^2}{2}{\cos }^{-1}\left(\frac{a-u}{a}\right)+C$
8. $\int \frac{du}{u\sqrt{2au-u^2}}=-\frac{\sqrt{2au-u^2}}{au}+C$

To Do

• Antiderivative DB search system.
• Find a use for the white-space on the right of the page.
• Integral Isolating System for easy reference/copying.