Table Of Integrals
Elementary Integrals
1. $\displaystyle\int \mathrm{d}u =u+C$
2. $\displaystyle\int a \mathrm{d}u = au+C$
3. $\displaystyle\int [f(u)+g(u)] = \displaystyle\int f(u) \mathrm{d}u +\displaystyle\int g(u) \mathrm{d}u$
4. $\displaystyle\int u^n \mathrm{d}u =\cfrac{u^{n+1}}{n+1}+C (n \neq -1)$
5. $\displaystyle\int \cfrac{\mathrm{d}u}{u} = \ln |u|+C$
Integrals Containing $a+bu$
6. $\displaystyle\int \cfrac{u\mathrm{d}u}{a+bu} =\cfrac{1}{b^2}[bu-a\ln|a+bu|]+C$
7. $\displaystyle\int \cfrac{u^2\mathrm{d}u}{a+bu} =\cfrac{1}{b^3}\left[\cfrac{1}{2}(a+bu)^2-2a(a+bu)+a^2\ln|a+bu| \right]+C$
8. $\displaystyle\int \cfrac{u\mathrm{d}u}{(a+bu)^2} = \cfrac{1}{b^2}\left[\cfrac{a}{a+bu}+\ln |a+bu|\right]+C$
9. $\displaystyle\int\cfrac{u^2\mathrm{d}u}{(a+bu)^2} =\cfrac{1}{b^3}\left[bu-\cfrac{a^2}{a+bu}-2a\ln|a+bu|\right]+C$
10. $\displaystyle\int \cfrac{u\mathrm{d}u}{(a+bu)^3}=\cfrac{1}{b^2}\left[\cfrac{a}{2(a+bu)^2}-\cfrac{1}{a+bu} \right]+C$
11. $\displaystyle\int \cfrac{mathrm{d}u}{u(a+bu)} =\cfrac{1}{a}\ln\left|\cfrac{u}{a+bu}\right|+C$
12. $\displaystyle\int \cfrac{\mathrm{d}u}{u^2(a+bu)}=-\cfrac{1}{au}+\cfrac{b}{a^2}\ln\left|\cfrac{a+bu}{u}\right| +C$
13. $\displaystyle\int \cfrac{\mathrm{d}u}{u(a+bu)^2} =\cfrac{1}{a(a+bu)}+\cfrac{1}{a^2}\ln \left|\cfrac{u}{a+bu}\right| +C$
14. $\displaystyle\int u\sqrt{a+bu}\mathrm{d}u=\cfrac{2}{15b^2}(3bu-2a)(a+bu)^{\cfrac{3}{2}}+C$
15. $\displaystyle\int u^2\sqrt{a+bu}\mathrm{d}u =\cfrac{2}{105b^3}(15b^2u^2-12abu+8a^2)(a+bu)^{\cfrac{3}{2}}+C$
16. $\displaystyle\int u^n\sqrt{a+bu}\mathrm{d}u =\cfrac{2u^n(a+bu)^{\cfrac{3}{2}}}{b(2n+3)}-\cfrac{2an}{b(2n+3)}\displaystyle\int u^{n-1}\sqrt{a+bu}\mathrm{d}u$
17. $\displaystyle\int\cfrac{u\mathrm{d}u}{\sqrt{a+bu}}=\cfrac{2}{3b^2}(bu-2a)\sqrt{a+bu}+C$
18. $\displaystyle\int \cfrac{u^2\mathrm{d}u}{\sqrt{a+bu}}=\cfrac{2}{15b^3}(3b^2u^2-4abu+8a^2)\sqrt{a+bu}+C$
Integrals Containing $\sqrt{a+bu}$
14. $\displaystyle\int u\sqrt{a+bu}\mathrm{d}u=\cfrac{2}{15b^2}(3bu-2a)(a+bu)^{\cfrac{3}{2}}+C$
15. $\displaystyle\int u^2\sqrt{a+bu}\mathrm{d}u =\cfrac{2}{105b^3}(15b^2u^2-12abu+8a^2)(a+bu)^{\cfrac{3}{2}}+C$
16. $\displaystyle\int u^n\sqrt{a+bu}\mathrm{d}u =\cfrac{2u^n(a+bu)^{\cfrac{3}{2}}}{b(2n+3)}-\cfrac{2an}{b(2n+3)}\displaystyle\int u^{n-1}\sqrt{a+bu}\mathrm{d}u$
17. $\displaystyle\int\cfrac{u\mathrm{d}u}{\sqrt{a+bu}}=\cfrac{2}{3b^2}(bu-2a)\sqrt{a+bu}+C$
18. $\displaystyle\int \cfrac{u^2\mathrm{d}u}{\sqrt{a+bu}}=\cfrac{2}{15b^3}(3b^2u^2-4abu+8a^2)\sqrt{a+bu}+C$
19. $\displaystyle\int\cfrac{u^n\mathrm{d}u}{\sqrt{a+bu}}=\cfrac{2u^n\sqrt{a+bu}}{b(2n+1)}-\cfrac{2an}{b(2n+1)}\displaystyle\int\cfrac{u^{n-1}\mathrm{d}u}{\sqrt{a+bu}}$
20. $\displaystyle\int\cfrac{\mathrm{d}u}{u\sqrt{a+bu}} = \begin{cases} \cfrac{1}{\sqrt{a}}\ln \left|\cfrac{\sqrt{a+bu}-\sqrt{a}}{\sqrt{a+bu}+\sqrt{a}}\right|+C & (a>0) \\ \cfrac{2}{\sqrt{-a}}\tan^{-1}\sqrt{\cfrac{a+bu}{-a}}+C & (a<0) \end{cases}$
21. $\displaystyle\int\cfrac{\mathrm{d}u}{u^n\sqrt{a+bu}}=-\cfrac{\sqrt{a+bu}}{a(n-1)u^{n-1}}-\cfrac{b(2n-3)}{2a(n-1)}\displaystyle\int\cfrac{\mathrm{d}u}{u^{n-1}\sqrt{a+bu}}$
22. $\displaystyle\int\cfrac{\sqrt{a+bu}\mathrm{d}u}{u}=2\sqrt{a+bu}+a\displaystyle\int\cfrac{\mathrm{d}u}{u\sqrt{a+bu}}$
23. $\displaystyle\int\cfrac{\sqrt{a+bu}\mathrm{d}u}{u^n}=-\cfrac{(a+bu)^{\cfrac{3}{2}}}{a(n-1)u^{n-1}}-\cfrac{b(2n-5)}{2a(n-1)}\displaystyle\int\cfrac{\sqrt{a+bu}\mathrm{d}u}{u^{n-1}}$
Integrals Containing $a^2\pm u^2$ $(a>0)$
24. $\displaystyle\int\cfrac{\mathrm{d}u}{a^2+u^2}=\cfrac{1}{a}\tan ^{-1}\cfrac{u}{a}+C$
25. $\displaystyle\int\cfrac{\mathrm{d}u}{a^2-u^2}=\cfrac{1}{2a}\ln\left|\cfrac{u+a}{u-a}\right| +C$
26. $\displaystyle\int\cfrac{\mathrm{d}u}{u^2-a^2}=\cfrac{1}{2a}\ln\left|\cfrac{u-a}{u+a}\right|+C$
No comments:
Post a Comment