本文复制粘贴自《积分公式大全》网络版本，作参考之用。
一、含有\(ax+b\)的积分
$$
1.\,\int\!\! \frac{dx}{ax+b}=\frac{1}{a}\ln \vert ax+b \vert +C
$$
$$
2.\,\int\!\! (ax+b)^\mu dx=\frac{1}{a(\mu+1)}(ax+b)^{\mu+1}+C \qquad
(\mu \neq -1)
$$
$$
3.\,\int\!\! \frac{x}{ax+b}dx=\frac{1}{a^2}(ax+b-b\ln\vert (x+b)
\vert ) +C
$$
$$
4.\,\int\!\!
\frac{x^2}{ax+b}dx=\frac{1}{a^3}\Big[\frac{1}{2}(ax+b)^2-2b(ax+b)+b^2\ln\vert
ax+b\vert\Big]+C
$$
$$
5.\,\int\!\! \frac{dx}{x(ax+b)}=-\frac{1}{b}\ln \Big\vert
\frac{ax+b}{x}\Big\vert +C
$$
$$
6.\,\int\!\!
\frac{dx}{x^2(ax+b)}=-\frac{1}{bx}+\frac{a}{b^2}\ln\Big\vert\frac{ax+b}{x}\Big\vert+C
$$
$$
7.\,\int\!\! \frac{x}{(ax+b)^2}dx=\frac{1}{a^2}\Big(\ln \vert ax+b
\vert + \frac{b}{ax+b}\Big)+C
$$
$$
8.\,\int\!\! \frac{x^2}{(ax+b)^2}dx=
\frac{1}{a^3}\Big(ax+b-2b\ln\vert ax+b \vert – \frac{b^2}{ax+b}\Big
) + C
$$
$$
9.\,\int\!\! \frac{dx}{x(a+b)^2}=\frac
{1}{b(ax+b)}-\frac{1}{b^2}\ln\Big\vert \frac{ax+b}{x}\Big\vert + C
$$
二、含有\(\sqrt{ax+b}\)的积分
$$
10.\,\int\!\! \sqrt{ax+b} dx= \frac{2}{3a}\sqrt{(ax+b)^3}+C
$$
$$
11.\,\int\!\! x
\sqrt{ax+b}dx=\frac{2}{15a^2}(3ax-2b)\sqrt{(ax+b)^3}+C
$$
$$
12.\,\int\!\! x^2
\sqrt{ax+b}dx=\frac{2}{105a^3}(15a^2x^2-12abx+8b^2)\sqrt{(ax+b)^3}+C
$$
$$
13.\,\int\!\! \frac{x}{\sqrt {ax+b}}dx=
\frac{2}{3a^2}(ax-2b)\sqrt{ax+b}+C
$$
$$
14.\,\int\!\! \frac{x^2}{\sqrt {ax+b}}dx=
\frac{2}{15a^2}(2a^2x^2-4abx+8b^2)\sqrt{ax+b}+C
$$
$$
15.\,\int\!\! \frac{dx}{x\sqrt{ax+b}}=\left \{
\begin{array}{cc}
\!\!\!\!\frac{1}{\sqrt{b}}\ln \Big \vert \frac{ \sqrt{ax+b} –
\sqrt{b}}{\sqrt{ax+b}+\sqrt{b}} \Big \vert + C & \textrm ( b>0) \\
\frac{2}{-\sqrt{-b}}\arctan \sqrt{\frac{ax+b}{-b}}+C & \textrm (b<0)
\end{array} \right.
$$
$$
16.\,\int\!\! \frac{dx}{x^2\sqrt{ax+b}} = – \frac{\sqrt{ax+b}}{bx} –
\frac{a}{2b}\int\!\! \frac{dx}{x\sqrt{ax+b}}
$$
$$
17.\,\int\!\! \frac{\sqrt{ax+b}}{x}dx = 2 \sqrt{ax+b}+b \int\!\!
\frac{dx}{x\sqrt{ax+b}}
$$
$$
18.\,\int\!\! \frac{\sqrt{ax+b}}{x^2}dx= – \frac{\sqrt{ax+b}}{x}+
\frac{a}{2}\int\!\! \frac{dx}{x\sqrt{ax+b}}
$$
三、含有\(x^2\pm a\)的积分
$$
19.\,\int\!\! \frac{dx}{x^2+a^2}= \frac{1}{a} \arctan \frac{x}{a} +C
$$
$$
20.\,\int\!\! \frac{dx}{(x^2+a^2)^n} =
\frac{x}{2(n-1)a^2(x^2+a^2)^{n-1}} + \frac {2n-3}{2(n-1)a^2}
\int\!\! \frac {dx}{(x^2+ a^2)^{n-1}}
$$
$$
21.\,\int\!\!\ \frac{dx}{x^2-a^2}= \frac{1}{2a} \ln \Big \vert
\frac{x-a}{x+a}\Big \vert + C
$$
四、含有\(ax^2 +b\)的积分
$$
22.\,\int\!\! \frac{dx}{ax^2+b} = \left \{
\begin{array}{cc}
\!\!\!\!\frac{1}{\sqrt{ab}}\arctan \sqrt {\frac {a}{b}}x +C & \textrm (b>0) \\
\frac{1}{2 \sqrt{-ab}} \ln\Big \vert \frac { \sqrt {a}x – \sqrt {-b}}{\sqrt{a}x+ \sqrt{-b}} \Big \vert + C
& \textrm (b<0)
\end{array} \right.
$$
$$
23.\,\int\!\! \frac{x} {ax^2 +b} dx = \frac{1}{2a} \ln \vert ax^2+b
\vert +C
$$
$$
24.\,\int\!\! \frac{x^2}{ax^2+b} dx= \frac{x}{a} – \frac{b}{a}
\int\!\! \frac{dx}{ax^2+b}
$$
$$
25.\,\int\!\! \frac {dx} { x(ax^2+b)} = \frac {1}{2b} \ln \frac
{x^2}{ \vert ax^2+b \vert } +C
$$
$$
26.\,\int\!\! \frac {dx}{x^2(ax^2+b)} = – \frac{1}{bx} – \frac{a}{b}
\int\!\! \frac {dx}{ax^2+b}
$$
$$
27.\,\int\!\! \frac{dx}{x^3(ax^2+b)} = \frac {a}{2b^2} \ln \frac {
\vert ax^2+b\vert }{x^2} – \frac{1}{2bx^2} + C
$$
$$
28.\,\int\!\! \frac{dx}{(ax^2+b)^2} = \frac{x}{2b(ax^2+b)} +
\frac{1}{2b} \int\!\! \frac {dx} { ax^2+b}
$$
五、含有\(ax^2+bx+c\)的积分
$$
29.\,\int\!\! \frac{dx}{ax^2+bx+c}dx= \left \{
\begin{array}{cc}
\!\!\! \frac{2}{\sqrt{4ac-b^2}} \arctan \frac {2ax+b}{\sqrt{4ac-b^2}}+C & \textrm (b^2<4ac) \\ \frac {1}{\sqrt {b^2-4ac}} \ln \Big \vert \frac {2ax+b-\sqrt{b^2-4ac}} {2ax+b+ \sqrt{b^2-4ac}} \Big \vert +C & \textrm (b^2>4ac)
\end{array} \right.
$$
$$
30.\,\int\!\! \frac{x}{ax^2+bx+c} dx = \frac{1}{2a}\ln \vert
ax^2+bx+c \vert – \frac{b}{2a} \int\!\! \frac {dx}{ax^2+bx+c}
$$
六、含有\(\sqrt{x^2+a^2} \quad (a>0)\)的积分
$$
31.\,\int\!\! \frac{dx}{\sqrt{x^2+a^2}} = arsh \frac{x}{a}+C1= \ln
(x+\sqrt{x^2+a^2})+C
$$
$$
32.\,\int\!\! \frac{dx}{\sqrt{(x^2+a^2)^3}}=
\frac{x}{a^2\sqrt{x^2+a^2}}+C
$$
$$
33.\,\int\!\! \frac{x}{\sqrt{x^2+a^2}}dx=\sqrt{x^2+a^2}+C
$$
$$
34.\,\int\!\! \frac{x}{\sqrt{(x^2+a^2)^3}}dx= –
\frac{1}{\sqrt{x^2+a^2}}+C
$$
$$
35.\,\int\!\! \frac{x^2}{\sqrt{x^2+a^2}}dx =
\frac{x}{2}\sqrt{x^2+a^2}-\frac{a^2}{2}\ln(x+\sqrt{x^2+a^2})+C
$$
$$
36.\,\int\!\! \frac{x^2}{\sqrt{(x^2+a^2)^3}}dx=
-\frac{x}{\sqrt{x^2+a^2}}+\ln (x+\sqrt{x^2+a^2})+C
$$
$$
37.\,\int\!\! \frac{dx}{x\sqrt{x^2+a^2}}= \frac{1}{a}\ln
\frac{\sqrt{x^2+a^2}-a}{\vert x \vert} +C
$$
$$
38.\,\int\!\! \frac{dx}{x^2\sqrt{x^2+a^2}}=
-\frac{\sqrt{x^2+a^2}}{a^2x}+C
$$
$$
39.\,\int\!\!
\sqrt{x^2+a^2}dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\ln(x+\sqrt{x^2+a^2})+C
$$
$$
40.\,\int\!\! \sqrt{(x^2+a^2)^3}dx =
\frac{x}{8}(2x^2+5a^2)\sqrt{x^2+a^2}+
\frac{3}{8}a^4\ln(x+\sqrt{x^2+a^2})+C
$$
$$
41.\,\int\!\! x\sqrt{x^2+a^2}dx= \frac{1}{3}\sqrt{(x^2+a^2)^3}+C
$$
$$
42.\,\int\!\! x^2
\sqrt{x^2+a^2}dx=\frac{x}{8}(2x^2+a^2)\sqrt{x^2+a^2}-\frac{a^4}{8}\ln(x+\sqrt{x^2+a^2})+C
$$
$$
43.\,\int\!\! \frac{\sqrt{x^2+a^2}}{x}dx=
\sqrt{x^2+a^2}+a\ln\frac{\sqrt{x^2+a^2}-a}{\vert x \vert} +C
$$
$$
44.\,\int\!\! \frac{\sqrt{x^2+a^2}}{x^2}dx=
-\frac{\sqrt{x^2+a^2}}{x} + \ln(x+\sqrt{x^2+a^2}) +C
$$
七、含有\(\sqrt{x^2-a^2} \quad (a>0) \)的积分
$$
45.\,\int\!\! \frac{dx}{\sqrt{x^2-a^2}}= \frac{x}{\vert x \vert}
arch \frac{\vert x \vert}{a}+C_1= ln \vert x+ \sqrt{x^2-a^2} \vert
+C
$$
$$
46.\,\int\!\! \frac{dx}{\sqrt{(x^2-a^2)^3}}= –
\frac{x}{a^2\sqrt{x^2-a^2}} +C
$$
$$
47.\,\int\!\! \frac{x}{\sqrt{x^2-a^2}}dx = \sqrt{x^2-a^2}+C
$$
$$
48.\,\int\!\! \frac{x}{\sqrt{(x^2-a^2)^3}}dx = –
\frac{1}{\sqrt{x^2-a^2}} +C
$$
$$
49.\,\int\!\! \frac{x^2}{\sqrt{x^2-a^2}}dx =\frac{x}{2}\sqrt{x^2 -a
^2} + \frac{a^2}{2} \ln \vert x+ \sqrt{x^2-a^2} \vert +C
$$
$$
50.\,\int\!\! \frac{x^2}{\sqrt{(x^2-a^2)^3}}dx =-\frac{x}{\sqrt{x^2
-a ^2}} + \ln \vert x+ \sqrt{x^2-a^2} \vert +C
$$
$$
51.\,\int\!\! \frac{dx}{x\sqrt{x^2-a^2}}= \frac{1}{a} \arccos
\frac{a}{\vert x \vert} +C
$$
$$
52.\,\int\!\! \frac{dx}{x^2\sqrt{x^2-a^2}} =\frac{ \sqrt
{x^2-a^2}}{a^2x}+C
$$
$$
53.\,\int\!\! \sqrt{x^2-a^2}dx = \frac{x}{2} \sqrt{x^2-a^2} –
\frac{a^2}{2} \ln \vert x+ \sqrt{x^2-a^2} \vert +C
$$
$$
54.\,\int\!\! \sqrt{(x^2-a^2)^3}dx
=\frac{x}{8}(2x^2-5a^2)\sqrt{x^2-a^2} + \frac{3}{8} a^4 \ln \vert x+
\sqrt{x^2-a^2} \vert +C
$$
$$
55.\,\int\!\! x \sqrt{x^2-a^2}dx= \frac{1}{3}\sqrt{(x^2-a^2)^3} +C
$$
$$
56.\,\int\!\! x^2 \sqrt{x^2-a^2} dx = \frac{x}{8}
(2x^2-a^2)\sqrt{x^2-a^2}-\frac{a^4}{8}\ln \vert x+ \sqrt{x^2-a^2}
\vert +C
$$
$$
57.\,\int\!\! \frac{\sqrt{x^2-a^2}}{x} dx= \sqrt{x^2-a^2}- \arccos
\frac{a}{\vert x \vert} +C
$$
$$
58.\,\int\!\! \frac{\sqrt{x^2-a^2}}{x^2} dx =
-\frac{\sqrt{x^2-a^2}}{x} + \ln \vert x + \sqrt{x^2-a^2} \vert +C
$$
八、含有\(\sqrt{a^2-x^2} \quad (a>0) \)的积分
$$
59.\,\int\!\! \frac{dx}{\sqrt{a^2-x^2}} =\arcsin \frac{x}{a} +C
$$
$$
60.\,\int\!\! \frac{dx}{\sqrt{(a^2-x^2)^3}}=
\frac{x}{a^2\sqrt{a^2-x^2}}+C
$$
$$
61.\,\int\!\! \frac{x}{\sqrt{a^2-x^2}} dx = – \sqrt{a^2-x^2}+C
$$
$$
62.\,\int\!\! \frac{x}{\sqrt{(a^2-x^2)^3}} dx = –
\frac{1}{\sqrt{a^2-x^2}} +C
$$
$$
63.\,\int\!\! \frac{x^2}{\sqrt{a^2-x^2}}dx =
-\frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\arcsin \frac{x}{a}+C
$$
$$
64.\,\int\!\! \frac{x^2}{\sqrt{(a^2-x^2)^3}}dx
=\frac{x}{\sqrt{a^2-x^2}}- \arcsin \frac{x}{a} +C
$$
$$
65.\,\int\!\!
\frac{dx}{x\sqrt{a^2-x^2}}=\frac{1}{a}\ln\frac{a-\sqrt{a^2-x^2}}{\vert
x \vert } +C
$$
$$
66.\,\int\!\! \frac{dx}{x^2\sqrt{a^2-x^2}}=- \frac{a^2-x^2}{a^2x}+C
$$
$$
67.\,\int\!\! \sqrt{a^2-x^2}dx= \frac{x}{2} \sqrt{a^2-x^2} +
\frac{a^2}{2}\arcsin \frac{x}{a}+C
$$
$$
68.\,\int\!\!
\sqrt{(a^2-x^2)^3}dx=\frac{x}{8}(5a^2-2x^2)\sqrt{a^2-x^2}+\frac{3}{8}a^4\arcsin\frac{x}{a}+C
$$
$$
69.\,\int\!\! x\sqrt{a^2-x^2}dx=-\frac{1}{3}\sqrt{(a^2-x^2)^3}+C
$$
$$
70.\,\int\!\!
x^2\sqrt{a^2-x^2}dx=\frac{x}{8}(2x^2-a^2)\sqrt{a^2-x^2}+\frac{a^4}{8}\arcsin\frac{x}{a}+C
$$
$$
71.\,\int\!\! \frac{\sqrt{a^2-x^2}}{x} dx =\sqrt{a^2-x^2}+a \ln
\frac{a-\sqrt{a^2-x^2}}{\vert x \vert}+C
$$
$$
72.\,\int\!\!
\frac{\sqrt{a^2-x^2}}{x^2}dx=-\frac{\sqrt{a^2-x^2}}{x}-\arcsin
\frac{x}{a}+C
$$
九、含有\(\sqrt{\pm ax^2+bx+c} \quad (a>0) \)的积分
$$
73.\,\int\!\! \frac{dx}{\sqrt{ax^2+bx+c}}=\frac{1}{\sqrt{a}}\ln
\vert 2ax+b+2\sqrt{a}\sqrt{ax^2+bx+c}\,\vert +C
$$
$$
74.\,\int\!\! \sqrt{ax^2+bx+c}\,dx=\frac{2ax+b}{4a}\sqrt{ax^2+bx+c}
$$
$$
\qquad
+\frac{4ac-b^2}{8\sqrt{a^3}}\ln\vert
2ax+b+2\sqrt{a}\sqrt{ax^2+bx+c}\vert +C
$$
$$
75.\,\int\!\! \frac{x}{\sqrt{ax^2 + bx + c}}dx = \frac{1}{a}\sqrt{
ax^2 + bx + c}
$$
$$
\qquad -\frac{b}{2\sqrt{a^3}}\ln \vert
2ax + b + 2 \sqrt{a}\sqrt{ax^2 + bx + c}\,\vert +C
$$
$$
76.\,\int\!\! \frac{dx}{\sqrt{c+bx-ax^2}}=
\frac{1}{\sqrt{a}}\arcsin\frac{2ax-b}{\sqrt{b^2+4ac}}+C
$$
$$
77.\,\int\!\!
\sqrt{c+bx-ax^2}\,dx=\frac{2ax-b}{4a}\sqrt{c+bx-ax^2}+\frac{b^2+4ac}{8\sqrt{a^3}}\arcsin
\frac{2ax-b}{\sqrt{b^2+4ac}}+C
$$
$$
78.\,\int\!\!
\frac{x}{\sqrt{c+bx-ax^2}}\,dx=-\frac{1}{a}\sqrt{c+bx-ax^2}+\frac{b}{2\sqrt{a^3}}\arcsin\frac{2ax-b}{\sqrt{b^2+4ac}}+C
$$
十、含有\(\sqrt{\pm \frac{x-a}{x+a}}\)或者\(\sqrt{(x-a)(b-x)}\)的积分
$$
79.\,\int\!\!
\sqrt{\frac{x-a}{x-b}}dx=(x-b)\sqrt{\frac{x-a}{x-b}}+(b-a)\ln(\sqrt{\vert
x-a\vert} + \sqrt{\vert x-b \vert }\,) +C
$$
$$
80.\,\int\!\!
\sqrt{\frac{x-a}{x-b}}\,dx=(x-b)\sqrt{\frac{x-a}{x-b}}+(b-a)\arcsin\sqrt{\frac{x-a}{b-a}}+C
$$
$$81.\,\int\!\!
\frac{dx}{\sqrt{(x-a)(x-b)}}=2\arcsin\sqrt{\frac{x-a}{b-a}}+C \quad
(a < b)
$$
$$
82.\,\int\!\!
\sqrt{(x-a)(b-x)}\,dx=\frac{2x-a-b}{4}\sqrt{(x-a)(b-x)}
$$
$$
\qquad
+\frac{(b-a)^2}{4}\arcsin\sqrt{\frac{x-a}{b-a}}+C \quad (a < b)
$$
十一、含有三角函数函数的积分
$$
83.\,\int\!\! \sin x dx = -\cos x +C
$$
$$
84.\,\int\!\! \cos x dx= \sin x +C
$$
$$
85.\,\int\!\! \tan x dx -\ln \vert \cos x \vert +C
$$
$$
86.\,\int\!\! ctgx dx =\ln \vert \sin x \vert +C
$$
$$
87.\,\int\!\! \sec x dx = \ln \vert \tan (\frac{ \pi}{4} +
\frac{x}{2}) \vert + C = \ln \vert \sec x + \tan x \vert +C
$$
$$
88.\,\int\!\! \csc x dx= \ln \vert \tan \frac{x}{2} \vert +C = \ln
\vert \csc x – ctg x \vert +C
$$
$$
89.\,\int\!\! \sec^2 x dx = \tan x +C
$$
$$
90.\,\int\!\! \csc ^2 x dx = – ctgx +C
$$
$$
91.\,\int\!\! \sec x \tan x dx = \sec x +C
$$
$$
92.\,\int\!\! \csc x dx ctgx dx = -\csc x +C
$$
$$
93.\,\int\!\! \sin ^2 x dx = \frac{x}{2}- \frac{1}{4}\sin 2x +C
$$
$$
94.\,\int\!\! \cos ^2 x dx = \frac{x}{2} + \frac{1}{4}\sin 2x +C
$$
$$
95.\,\int\!\! \sin ^n x dx = – \frac{1}{n}\sin ^{n-1}x \cos x +
\frac{n-1}{n} \int\!\! \sin ^{n-2}dx
$$
$$
96.\,\int\!\! \cos ^ n x dx = \frac{1}{n}\cos^{ n-1}x \sin x +
\frac{n-1}{n} \int\!\! \cos^{n-2} x dx
$$
$$
97.\,\int\!\! \frac{dx}{\sin ^ n x} = – \frac{1}{n-1} . \frac{\cos
x}{\sin ^{n-1}x}+\frac{n-2}{n-1} \int\!\! \frac{dx}{\sin^{n-2}x}
$$
$$
98.\,\int\!\! \frac{dx}{\cos ^n x}= \frac{1}{n-1}.\frac{\sin x}{\cos
^{n-1}x}+\frac{n-2}{n-1}\int\!\! \frac{dx}{\cos x^{n-2}x}
$$
$$
99.\,\int\!\! \cos ^ m \sin ^n x dx =\frac{1}{m+n}\cos^{m-1}x \sin
^{n+1}x + \frac{m-1}{m+n} \int\!\! \cos ^ {m-2} x \sin ^n x dx
$$
$$
\qquad = -\frac{1}{m+1}\cos ^{m+1}x \sin ^{n-1}x + \frac{n-1}{m+n}
\int\!\! \cos ^m x \sin ^{n-2} x dx
$$
$$
100.\,\int\!\! \sin ax \cos bx dx = – \frac{1}{2(a+b)}\cos (a+b)x –
\frac{1}{2(a-b)} \cos (a-b)x +C
$$
$$
101.\,\int\!\! \sin ax \sin bx dx = – \frac{1}{2(a+b)} \sin (a+b) x
+ \frac{1}{2(a-b)} \sin (a-b)x +C
$$
$$
102.\,\int\!\! \cos ax \cos bx dx =\frac{1}{2(a+b)} \sin (a+b)x +
\frac{1}{2(a-b)} \sin (a-b)x +C
$$
$$
103.\,\int\!\! \frac{dx}{a+b\sin x} =
\frac{2}{\sqrt{a^2-b^2}}\arctan \frac{\arctan
\frac{x}{2}+b}{\sqrt{a^2-b^2}} +C \qquad ( a^2 > b^2 )
$$
$$
104.\,\int\!\! \frac{dx}{a+b \sin x} = \frac{1}{\sqrt{b^2-a^2}} \ln
\Bigg \vert \frac{\arctan \frac{x}{2}+b – \sqrt{b^2-a^2}}{ \arctan
\frac {x}{2}+b+ \sqrt {b^2-a^2}} \Bigg \vert +C \qquad (a^2 < b^2)
$$
$$
105.\,\int\!\! \frac{dx}{a+b \cos x} = \frac{2}{a+b} \sqrt{
\frac{a+b}{a-b}} \arctan \Bigg ( \sqrt { \frac{a-b}{a+b}} \tan
\frac{x}{2} \Bigg ) +C \qquad (a^2>b^2)
$$
$$
106.\,\int\!\! \frac{dx}{a+b \cos x}= \frac{1}{a+b}\sqrt{
\frac{a+b}{b-a}} \ln \Bigg \vert \frac{\tan \frac{x}{2}+ \sqrt
{\frac{a+b}{b-a}}}{\tan \frac{x}{2}- \sqrt{\frac{a+b}{b-a}}} \Bigg
\vert +C \qquad (a^2 < b^2)
$$
$$
107.\,\int\!\! \frac{dx}{a^2\cos ^2x + b^2 \sin ^2 x}= \frac{1}{ab}
\arctan (\frac{b}{a}\tan x ) +C
$$
$$
108.\,\int\!\! \frac{dx}{a^2 \cos ^2x -b^2 \sin ^2 x} =
\frac{1}{2ab} \ln \Big \vert \frac{b \tan x +a }{b \tan x -a} \Big
\vert +C
$$
$$
109.\,\int\!\! x \sin ax dx = \frac{1}{a^2} \sin ax – \frac{1}{a} x
\cos ax +C
$$
$$
110.\,\int\!\! x^2 \sin ax dx = -\frac{1}{a}x^2 \cos ax +
\frac{2}{a^2}x\sin ax + \frac{2}{a^3}\cos ax +C
$$
$$
111.\,\int\!\! x \cos ax dx = \frac{1}{a^2} \cos ax + \frac{1}{a}x
\sin ax +C
$$
$$
112.\,\int\!\! x^2 \cos ax dx = \frac{1}{a} x^2 \sin ax +
\frac{2}{a^2}x \cos ax – \frac{2}{a^3}\sin ax +C
$$
十二、含有反三角函数的积分,其中\(a > 0 \)
$$
113.\,\int\!\! \arcsin \frac{x}{a} dx = x \arcsin \frac{x}{a}+
\sqrt{a^2-x^2} +C
$$
$$
114.\,\int\!\! x\arcsin \frac{x}{a} dx = \Big (
\frac{x^2}{2}-\frac{a^2}{4}\Big )\arcsin \frac{x}{a} +
\frac{x}{4}\sqrt {a^2-x^2} +C
$$
$$
115.\,\int\!\! x^2 \arcsin \frac{x}{a}dx= \frac{x^3}{3} \arcsin
\frac{x}{a} + \frac{1}{9}(x^2+2a^2)\sqrt{a^2-x^2}+C
$$
$$
116.\,\int\!\! \arccos \frac{x}{a}dx= x \arccos \frac{x}{a}-\sqrt
{a^2-x^2} +C
$$
$$
117.\,\int\!\! x \arccos \frac{x}{a} dx = \Big (
\frac{x^2}{2}-\frac{a^2}{4}\Big) \arccos \frac{x}{4} –
\frac{x}{4}\sqrt {a^2 -x^2} +C
$$
$$
118.\,\int\!\! x^2\arccos \frac{x}{a} dx = \frac{x^3}{a} \arccos
\frac{x}{a} – \frac{1}{9}(x^2 +2a^2)\sqrt{a^2-x^2}+C
$$
$$
119.\,\int\!\! \arctan \frac{x}{a} dx =x \arctan \frac{x}{a} –
\frac{a}{2}\ln(a^2+x^2)+C
$$
$$
120.\,\int\!\! x\arctan \frac{x}{a}dx = \frac{1}{2}(a^2+x^2)\arctan
\frac{x}{a}-\frac{a}{2}x+C
$$
$$
121.\,\int\!\! x^2 \arctan \frac{x}{a}dx = \frac{x^3}{3}\arctan
\frac{x}{a} – \frac{a}{6}x^2+ \frac{a^3}{6}\ln (a^2+x^2)+C
$$
十三、含有指数函数的积分，其中\(a>0\)
$$
122.\,\int\!\! a^x dx = \frac{1}{\ln a} a^x +C
$$
$$
123.\,\int\!\! e^{ax}dx = \frac{1}{a} e ^{ax}+C
$$
$$
124.\,\int\!\! xe^{ax}dx=\frac{1}{a^2}(ax-1)e^{ax}+c
$$
$$
125.\,\int\!\! x^n e^{ax}dx=\frac{1}{a}x^n e^{ax}-
\frac{n}{a}\int\!\! x^{n-1} e^{ax}dx
$$
$$
126.\,\int\!\! xa^x dx= \frac{x}{\ln a}a^x – \frac{1}{(\ln
a)^2}a^x+C
$$
$$
127.\,\int\!\! x^n a^x dx=\frac{1}{\ln a}x^na^x – \frac{n}{\ln
a}\int\!\! x^{n-1} a^x dx
$$
$$
128.\,\int\!\! e^{ax} \sin bx dx = \frac{1}{a^2+b^2}e^{ax}(a\sin bx
-b \cos bx)+C
$$
$$
129.\,\int\!\! e^{ax}\cos bx dx = \frac{1}{a^2+b^2} e^{ax} (b \sin
bx + a \cos bx) +C
$$
$$
130.\,\int\!\! e^{ax}\sin ^n bx dx = \frac{1}{a^2+b^2n^2}e^{ax}\sin
^{n-1}bx (a \sin bx – nb \cos bx)
$$
$$
\qquad \qquad \qquad \qquad \qquad+
\frac{n(n-1)b^2}{a^2+b^2n^2}\int\!\! e^{ax}\sin ^{n-2}bx dx
$$
$$
131.\,\int\!\! e^{ax} \cos ^n bx dx = \frac{1}{a^2+b^2n^2} e^{ax}
\cos ^{n-1} bx (a\cos bx + nb \sin bx)
$$
$$
\qquad \qquad \qquad \qquad \qquad + \frac{n(n-1)b^2}{a^2+b^2n^2}
\int\!\! e^{ax} \cos ^{n-2}bx dx
$$
十四、含有对数函数的积分
$$
132. \, \int\!\!\ln x dx=x\ln x -x +C
$$
$$
133.\,\int\!\! \frac {dx}{x\ln x}= \ln \vert \ln x \vert +C
$$
$$
134. \,\int\!\! x^n\ln x dx=\frac{1}{n+1}x^{n+1}(\ln x –
\frac{1}{n+1})+C
$$
$$
135. \,\int\!\! (\ln x)^n dx=x(\ln x)^n – n \int\!\!(\ln x)^{n-1} dx
$$
$$
136. \,\int\!\! x^m (\ln x)^n dx=\frac{1}{m+1}x^{m+1}(\ln x)^n –
\frac {n}{m+1} \int\!\! x^m (\ln x) ^{n-1} dx
$$
十五、含有双曲函数的积分
$$
137. \,\int\!\! \sinh x dx = \cosh x +C
$$
$$
138. \,\int\!\! \cosh x dx = \sinh x +C
$$
$$
139. \,\int\!\! th x dx =\ln \cosh x +C
$$
$$
140. \,\int\!\! \sinh ^2 x dx = -\frac{x}{2} + \frac{1}{4} \sinh 2x
+C
$$
$$
141. \,\int\!\! \cosh^2 x dx = \frac{x}{2}+\frac{1}{4} \sinh 2x +C
$$
十六、定积分
$$
142. \,\int\!\! ^{\pi}_{-\pi} \cos nx dx = \int\!\! ^{\pi}_{\pi}
\sin nx dx=0
$$
$$
143. \,\int\!\! ^ {\pi} _{-\pi} \cos mx \sin nx dx =0
$$
$$
144. \,\int\!\! ^{\pi} _{-\pi} \cos mx \cos nx dx =\left \{
\begin{array}{cc} 0, & m \neq n \\ \pi, & m=n \end{array} \right.
$$
$$
145. \,\int\!\! ^\pi _{-\pi} \sin mx \sin nx dx = \left \{
\begin{array}{cc} 0, & m \neq n \\ \pi, & m=n \end{array} \right.
$$
$$
146. \,\int\!\! ^\pi _0 \sin mx \sin nx dx= \int\!\! ^\pi _0 \cos mx
\cos nx dx = \left \{
\begin{array}{cc} 0, & m \neq n \\ \pi/2, & m=n \end{array} \right.
$$
$$
147. I_n =\,\int\!\! ^{\frac{\pi}{2}} _0 \sin ^n x dx = \int\!\!
^{\frac{\pi}{2}} _0 \cos ^n x dx;\qquad I_n= \frac{n-1}{n} I _{n-2}$$
$$I_n= \frac{n-1}{n}.\frac{n-3}{n-2}.\dots \frac{4}{5} .\frac{2}{3}(n为大于1的正奇数),I_1=1
$$
$$
I_n= \frac{n-1}{n} . \frac{n-3}{n-2} . \dots \frac{3}{4} .\frac{1}{2}. \frac{\pi}{2}
(n为正偶数),I_0= \frac{\pi}{2}$$