1. 含有\(ax+b\)的积分
\[ \int \frac{dx} {ax+b} =\frac{1} {a} \ln|ax+b|+C \]
\[ \int(ax+b)^\mu dx=\frac{1} {a(\mu+1)} (ax+b)^{\mu+1} +C (\mu\ne−1) \]
\[ \int \frac{x} {ax+b} dx=\frac{1} {a^2} (ax+b−b\ln|(x+b)|)+C \]
\[ \int \frac{x^{2} } {ax+b} dx=\frac{1} {a^{3} } [\frac{1} {2} (ax+b)^{2} − 2b(ax+b)+b^{2} \ln|ax+b|]+C \]
\[ \int \frac{dx} {x(ax+b)} =−\frac{1} {b} \ln∣\frac{ax+b} {x} ∣+C \]
\[ \int \frac{dx} {x^2(ax+b)} = −\frac{1} {bx} +\frac{a} {b^2} \ln∣\frac{ax+b} {x} |+C \]
\[ \int \frac{x} {(ax+b)^2} dx=\frac{1} {a^2} (\ln|ax+b|+\frac{b} {ax+b} )+C \]
\[ \int \frac{x^2} {(ax+b)^2} dx=\frac{1} {a^3} (ax+b−2b\ln|ax+b|−\frac{b^2} {ax+b} )+C \]
\[ \int \frac{dx} {x(a+b)^2} =\frac{1} {b(ax+b)} −\frac{1} {b^2} \ln∣\frac{ax+b} {x} ∣+C \]
2. 含有$ $ 的积分
\[ \int \sqrt{ax+b} dx=\frac{2} {3a} \sqrt{(ax+b)^3} C \]
\[ \int x\sqrt{ax+b} dx=\frac{2} {15a^2} (3ax−2b)\sqrt{(ax+b)^3} +C \]
\[ \int x^2 \sqrt{ax+b} dx=\frac{2} {105a^3} (15a^2 x^2 −12abx+8b^2) \sqrt{(ax+b)^3} +C \]
\[ \int \frac{x} {\sqrt{ax+b} } dx=\frac{2} {3a^2} (ax−2b) \sqrt{ax+b} +C \]
\[ \int \frac{x^2} {\sqrt{ax+b} } dx=\frac{2} {15a^2} (2a^2 x^2 −4abx+8b^2) \sqrt{ax+b} +C \]
\[ \int \frac{d x} {x \sqrt{a x+b} } = \left \{ \begin{array} {ll} \frac{1} {\sqrt{b} } \ln \left| \frac{\sqrt{a x+b} - \sqrt{b} } {\sqrt{a x+b} +\sqrt{b} } \right|+C & (b>0) \\ \frac{2} {-\sqrt{-b} } \arctan \sqrt{\frac{a x+b} {-b} } +C & (b<0) \end{array} \right . \]
\[ \int \frac{d x} {x^{2} \sqrt{a x+b} } =-\frac{\sqrt{a x+b} } {b x} -\frac{a} {2 b} \int \frac{d x} {x \sqrt{a x+b} } \]
\[ \int \frac{\sqrt{a x+b} } {x} d x=2 \sqrt{a x+b} +b \int \frac{d x} {x \sqrt{a x+b} } \]
\[ \int \frac{\sqrt{a x+b} } {x^{2} } d x=-\frac{\sqrt{a x+b} } {x} +\frac{a} {2} \int \frac{d x} {x \sqrt{a x+b} } \]
3. 含有\(x^2\pm a\) 的积分
\[ \int \frac{d x} {x^{2} +a^{2} } =\frac{1} {a} \arctan \frac{x} {a} +C \]
\[ \int \frac{d x} {\left(x^{2} +a^{2} \right)^{n} } =\frac{x} {2(n-1) a^{2} \left(x^{2} +a^{2} \right)^{n-1} } +\frac{2 n-3} {2(n-1) a^{2} } \int \frac{d x} {\left(x^{2} +a^{2} \right)^{n-1} } \]
\[ \int \frac{d x} {x^{2} -a^{2} } =\frac{1} {2 a} \ln \left|\frac{x-a} {x+a} \right|+C \]
4. 含有\(ax^2+b\) 的积分
\[ \int \frac{d x} {a x^{2} +b} =\left\{\begin{array} {ll} \frac{1} {\sqrt{a b} } \arctan \sqrt{\frac{a} {b} } x+C & (b>0) \\ \frac{1} {2 \sqrt{-a b} } \ln \left|\frac{\sqrt{a} x-\sqrt{-b} } {\sqrt{a x+\sqrt{-b} } } \right|+C & (b<0)\end{array} \right. \]
\[ \int \frac{x} {a x^{2} +b} d x=\frac{1} {2 a} \ln \left|a x^{2} +b\right|+C \]
\[ \int \frac{x^{2} } {a x^{2} +b} d x=\frac{x} {a} -\frac{b} {a} \int \frac{d x} {a x^{2} +b} \]
\[ \int \frac{d x} {x\left(a x^{2} +b\right)} =\frac{1} {2 b} \ln \frac{x^{2} } {\left|a x^{2} +b\right|} +C \]
\[ \int \frac{d x} {x^{2} \left(a x^{2} +b\right)} =-\frac{1} {b x} -\frac{a} {b} \int \frac{d x} {a x^{2} +b} \]
\[ \int \frac{d x} {x^{3} \left(a x^{2} +b\right)} =\frac{a} {2 b^{2} } \ln \frac{\left|a x^{2} +b\right|} {x^{2} } -\frac{1} {2 b x^{2} } +C \]
\[ \int \frac{d x} {\left(a x^{2} +b\right)^{2} } =\frac{x} {2 b\left(a x^{2} +b\right)} +\frac{1} {2 b} \int \frac{d x} {a x^{2} +b} \]
5. 含有\(ax^2+bx+c\) 的积分
\[ \int \frac{d x} {a x^{2} +b x+c} d x=\left\{\begin{array} {ll} \frac{2} {\sqrt{4 a c-b^{2} } } \arctan \frac{2 a x+b} {\sqrt{4 a c-b^{2} } } +C & \left(b^{2} <4 a c\right) \\ \frac{1} {\sqrt{b^{2} -4 a c} } \ln \left|\frac{2 a x+b-\sqrt{b^{2} -4 a c} } {2 a x+b+\sqrt{b^{2} -4 a c} } \right|+C & \left(b^{2} >4 a c\right)\end{array} \right. \]
\[ \int \frac{x} {a x^{2} +b x+c} d x=\frac{1} {2 a} \ln \left|a x^{2} +b x+c\right|-\frac{b} {2 a} \int \frac{d x} {a x^{2} +b x+c} \]
6. 含有\(\sqrt{x^2+a^2} \quad\left(a>0\right)\) 的积分
\[ \int \frac{d x} {\sqrt{x^{2} +a^{2} } } =\operatorname{arsh} \frac{x} {a} +C 1=\ln (x+\sqrt{x^{2} +a^{2} } )+C \]
\[ \int \frac{d x} {\sqrt{\left(x^{2} +a^{2} \right)^{3} } } =\frac{x} {a^{2} \sqrt{x^{2} +a^{2} } } +C \]
\[ \int \frac{x} {\sqrt{x^{2} +a^{2} } } d x=\sqrt{x^{2} +a^{2} } +C \]
\[ \int \frac{x} {\sqrt{\left(x^{2} +a^{2} \right)^{3} } } d x=-\frac{1} {\sqrt{x^{2} +a^{2} } } +C \]
\[ \int \frac{x^{2} } {\sqrt{x^{2} +a^{2} } } d x=\frac{x} {2} \sqrt{x^{2} +a^{2} } -\frac{a^{2} } {2} \ln (x+\sqrt{x^{2} +a^{2} } )+C \]
\[ \int \frac{x^{2} } {\sqrt{\left(x^{2} +a^{2} \right)^{3} } } d x=-\frac{x} {\sqrt{x^{2} +a^{2} } } +\ln (x+\sqrt{x^{2} +a^{2} } )+C \]
\[ \int \frac{d x} {x \sqrt{x^{2} +a^{2} } } =\frac{1} {a} \ln \frac{\sqrt{x^{2} +a^{2} } -a} {|x|} +C \]
\[ \int \frac{d x} {x^{2} \sqrt{x^{2} +a^{2} } } =-\frac{\sqrt{x^{2} +a^{2} } } {a^{2} x} +C \]
\[ \int \sqrt{x^{2} +a^{2} } d x=\frac{x} {2} \sqrt{x^{2} +a^{2} } +\frac{a^{2} } {2} \ln (x+\sqrt{x^{2} +a^{2} } )+C \]
\[ \int \sqrt{\left(x^{2} +a^{2} \right)^{3} } d x=\frac{x} {8} \left(2 x^{2} +5 a^{2} \right) \sqrt{x^{2} +a^{2} } +\frac{3} {8} a^{4} \ln (x+\sqrt{x^{2} +a^{2} } )+C \]
\[ \int x \sqrt{x^{2} +a^{2} } d x=\frac{1} {3} \sqrt{\left(x^{2} +a^{2} \right)^{3} } +C \]
\[ \int x^{2} \sqrt{x^{2} +a^{2} } d x=\frac{x} {8} \left(2 x^{2} +a^{2} \right) \sqrt{x^{2} +a^{2} } -\frac{a^{4} } {8} \ln (x+\sqrt{x^{2} +a^{2} } ) +C \]
\[ \int \frac{\sqrt{x^{2} +a^{2} } } {x} d x=\sqrt{x^{2} +a^{2} } +a \ln \frac{\sqrt{x^{2} +a^{2} } -a} {|x|} +C \]
\[ \int \frac{\sqrt{x^{2} +a^{2} } } {x^{2} } d x=-\frac{\sqrt{x^{2} +a^{2} } } {x} +\ln (x+\sqrt{x^{2} + a^{2} } ) + C \]
7. 含有\(\sqrt{x^2-a^2} \quad(a>0)\) 的积分
\[ \int \frac{d x} {\sqrt{x^{2} -a^{2} } } =\frac{x} {|x|} \operatorname{arch} \frac{|x|} {a} +C_{1} =\ln |x+\sqrt{x^{2} -a^{2} } |+c \]
\[ \int \frac{d x} {\sqrt{\left(x^{2} -a^{2} \right)^{3} } } =-\frac{x} {a^{2} \sqrt{x^{2} -a^{2} } } +C \]
\[ \int \frac{x} {\sqrt{x^{2} -a^{2} } } d x=\sqrt{x^{2} -a^{2} } +C \]
\[ \int \frac{x} {\sqrt{\left(x^{2} -a^{2} \right)^{3} } } d x=-\frac{1} {\sqrt{x^{2} -a^{2} } } +C \]
\[ \int \frac{x^{2} } {\sqrt{x^{2} -a^{2} } } d x=\frac{x} {2} \sqrt{x^{2} -a^{2} } +\frac{a^{2} } {2} \ln |x+\sqrt{x^{2} -a^{2} } |+C \]
\[ \int \frac{x^{2} } {\sqrt{\left(x^{2} -a^{2} \right)^{3} } } d x=-\frac{x} {\sqrt{x^{2} -a^{2} } } +\ln |x+\sqrt{x^{2} -a^{2} } |+C \]
\[ \int \frac{d x} {x \sqrt{x^{2} -a^{2} } } =\frac{1} {a} \arccos \frac{a} {|x|} +C \]
\[ \int \frac{d x} {x^{2} \sqrt{x^{2} -a^{2} } } =\frac{\sqrt{x^{2} -a^{2} } } {a^{2} x} +C \]
\[ \int \sqrt{x^{2} -a^{2} } d x=\frac{x} {2} \sqrt{x^{2} -a^{2} } -\frac{a^{2} } {2} \ln | x+\sqrt{x^{2} -a^{2} |} +C \]
\[ \int \sqrt{\left(x^{2} -a^{2} \right)^{3} } d x=\frac{x} {8} \left(2 x^{2} -5 a^{2} \right) \sqrt{x^{2} -a^{2} } +\frac{3} {8} a^{4} \ln |x+\sqrt{x^{2} -a^{2} } |+C \]
\[ \int x \sqrt{x^{2} -a^{2} } d x=\frac{1} {3} \sqrt{\left(x^{2} -a^{2} \right)^{3} } +C \]
\[ \int x^{2} \sqrt{x^{2} -a^{2} } d x=\frac{x} {8} \left(2 x^{2} -a^{2} \right) \sqrt{x^{2} -a^{2} } -\frac{a^{4} } {8} \ln |x+\sqrt{x^{2} -a^{2} } |+C \]
\[ \int \frac{\sqrt{x^{2} -a^{2} } } {x} d x=\sqrt{x^{2} -a^{2} } -\arccos \frac{a} {|x|} +C \]
\[ \int \frac{\sqrt{x^{2} -a^{2} } } {x^{2} } d x=-\frac{\sqrt{x^{2} -a^{2} } } {x} +\ln |x+\sqrt{x^{2} -a^{2} } |+C \]
8. 含有\(\sqrt{a^2-x^2} \quad(a>0)\) 的积分
\[ \int \frac{d x} {\sqrt{a^{2} -x^{2} } } =\arcsin \frac{x} {a} +C \]
\[ \int \frac{d x} {\sqrt{\left(a^{2} -x^{2} \right)^{3} } } =\frac{x} {a^{2} \sqrt{a^{2} -x^{2} } } +C \]
\[ \int \frac{x} {\sqrt{a^{2} -x^{2} } } d x=-\sqrt{a^{2} -x^{2} } +C \]
\[ \int \frac{x} {\sqrt{\left(a^{2} -x^{2} \right)^{3} } } d x=-\frac{1} {\sqrt{a^{2} -x^{2} } } +C \]
\[ \int \frac{x^{2} } {\sqrt{a^{2} -x^{2} } } d x=-\frac{x} {2} \sqrt{a^{2} -x^{2} } +\frac{a^{2} } {2} \arcsin \frac{x} {a} +C \]
\[ \int \frac{x^{2} } {\sqrt{\left(a^{2} -x^{2} \right)^{3} } } d x=\frac{x} {\sqrt{a^{2} -x^{2} } } -\arcsin \frac{x} {a} +C \]
\[ \int \frac{d x} {x \sqrt{a^{2} -x^{2} } } =\frac{1} {a} \ln \frac{a-\sqrt{a^{2} -x^{2} } } {|x|} +C \]
\[ \int \frac{d x} {x^{2} \sqrt{a^{2} -x^{2} } } =-\frac{a^{2} -x^{2} } {a^{2} x} +C \]
\[ \int \sqrt{a^{2} -x^{2} } d x=\frac{x} {2} \sqrt{a^{2} -x^{2} } +\frac{a^{2} } {2} \arcsin \frac{x} {a} +C \]
\[ \int \sqrt{\left(a^{2} -x^{2} \right)^{3} } d x=\frac{x} {8} \left(5 a^{2} -2 x^{2} \right) \sqrt{a^{2} -x^{2} } +\frac{3} {8} a^{4} \text { arcsin } \frac{x} {a} +C \]
\[ \int x \sqrt{a^{2} -x^{2} } d x=-\frac{1} {3} \sqrt{\left(a^{2} -x^{2} \right)^{3} } +C \]
\[ \int x^{2} \sqrt{a^{2} -x^{2} } d x=\frac{x} {8} \left(2 x^{2} -a^{2} \right) \sqrt{a^{2} -x^{2} } +\frac{a^{4} } {8} \arcsin \frac{x} {a} +C \]
\[ \int \frac{\sqrt{a^{2} -x^{2} } } {x} d x=\sqrt{a^{2} -x^{2} } +a \ln \frac{a-\sqrt{a^{2} -x^{2} } } {|x|} +C \]
\[ \int \frac{\sqrt{a^{2} -x^{2} } } {x^{2} } d x=-\frac{\sqrt{a^{2} -x^{2} } } {x} -\arcsin \frac{x} {a} +C \]
9. 含有 \(\sqrt{\pm ax^2+bx+c} \quad(a>0)\)的积分
\[ \int \frac{d x} {\sqrt{a x^{2} +b x+c} } =\frac{1} {\sqrt{a} } \ln |2 a x+b+2 \sqrt{a} \sqrt{a x^{2} +b x+c} |+C \]
\[ \int \sqrt{a x^{2} +b x+c} d x=\frac{2 a x+b} {4 a} \sqrt{a x^{2} +b x+c} +\frac{4 a c-b^{2} } {8 \sqrt{a^{3} } } \ln |2 a x+b+2 \sqrt{a} \sqrt{a x^{2} +b x+c} |+C \]
\[ \int \frac{x} {\sqrt{a x^{2} +b x+c} } d x=\frac{1} {a} \sqrt{a x^{2} +b x+c} -\frac{b} {2 \sqrt{a^{3} } } \ln |2 a x+b+2 \sqrt{a} \sqrt{a x^{2} +b x+c} |+C \]
\[ \int \frac{d x} {\sqrt{c+b x-a x^{2} } } =\frac{1} {\sqrt{a} } \arcsin \frac{2 a x-b} {\sqrt{b^{2} +4 a c} } +C \]
\[ \int \sqrt{c+b x-a x^{2} } d x=\frac{2 a x-b} {4 a} \sqrt{c+b x-a x^{2} } +\frac{b^{2} +4 a c} {8 \sqrt{a^{3} } } \arcsin \frac{2 a x-b} {\sqrt{b^{2} +4 a c} } +C \]
\[ \int \frac{x} {\sqrt{c+b x-a x^{2} } } d x=-\frac{1} {a} \sqrt{c+b x-a x^{2} } +\frac{b} {2 \sqrt{a^{3} } } \arcsin \frac{2 a x-b} {\sqrt{b^{2} +4 a c} } +C \]
10. 含有$ \(或者\) $的积分
\[ \int \sqrt{\frac{x-a} {x-b} } d x=(x-b) \sqrt{\frac{x-a} {x-b} } +(b-a) \ln (\sqrt{|x-a|} +\sqrt{|x-b|} )+C \]
\[ \int \sqrt{\frac{x-a} {x-b} } d x=(x-b) \sqrt{\frac{x-a} {x-b} } +(b-a) \arcsin \sqrt{\frac{x-a} {b-a} } +C \]
\[ \int \frac{d x} {\sqrt{(x-a)(x-b)} } =2 \arcsin \sqrt{\frac{x-a} {b-a} } +C \quad(a<b) \]
\[ \int \sqrt{(x-a)(b-x)} d x=\frac{2 x-a-b} {4} \sqrt{(x-a)(b-x)} +\frac{(b-a)^{2} } {4} \arcsin \sqrt{\frac{x-a} {b-a} } +C \quad(a<b) \]
11. 含有三角函数函数的积分
\[ \int \sin x d x=-\cos x+C \]
\[ \int \cos x d x=\sin x+C \]
\[ \int \tan x d x-\ln |\cos x|+C \]
\[ \int \operatorname{ctg} x d x=\ln |\sin x|+C \]
\[ \int \sec x d x=\ln \left|\tan \left(\frac{\pi} {4} +\frac{x} {2} \right)\right|+C=\ln |\sec x+\tan x|+C \]
\[ \int \csc xdx=\ln | \tan \frac{x} {2} | +C=\ln |\csc x-\operatorname{ctg} x|+C \]
\[ \int \sec ^ {2} xdx = \tan x + C \]
\[ \int \csc ^ {2} xdx = - \operatorname{ctg} x + C \]
\[ \int \sec x \tan x d x=\sec x+C \]
\[ \int \csc x d x \operatorname{ctg} x d x=-\csc x+C \]
\[ \int \sin ^{2} x d x=\frac{x} {2} -\frac{1} {4} \sin 2 x+C \]
\[ \int \cos ^{2} x d x=\frac{x} {2} +\frac{1} {4} \sin 2 x+C \]
\[ \int \sin ^{n} x d x=-\frac{1} {n} \sin ^{n-1} x \cos x+\frac{n-1} {n} \int \sin ^{n-2} d x \]
\[ \int \cos ^{n} x d x=\frac{1} {n} \cos ^{n-1} x \sin x+\frac{n-1} {n} \int \cos ^{n-2} x d x \]
\[ \int \frac{d x} {\sin ^{n} x} =-\frac{1} {n-1} \cdot \frac{\cos x} {\sin ^{n-1} x} +\frac{n-2} {n-1} \int \frac{d x} {\sin ^{n-2} x} \]
\[ \int \frac{d x} {\cos ^{n} x} =\frac{1} {n-1} \cdot \frac{\sin x} {\cos ^{n-1} x} +\frac{n-2} {n-1} \int \frac{d x} {\cos x^{n-2} x} \]
\[ \int \cos ^{m} \sin ^{n} x d x=\frac{1} {m+n} \cos ^{m-1} x \sin ^{n+1} x+\frac{m-1} {m+n} \int \cos ^{m-2} x \sin ^{n} x d x \\ = - \frac {1} {m+1} \cos ^ {m+1} x \sin ^ {n-1} x + \frac {n-1} {m+n} \int \cos ^ {m} x \sin ^ {n-2} xdx \]
\[ \int \sin a x \cos b x d x=-\frac{1} {2(a+b)} \cos (a+b) x-\frac{1} {2(a-b)} \cos (a-b)x + C \]
\[ \int \sin a x \sin b x d x=-\frac{1} {2(a+b)} \sin (a+b) x+\frac{1} {2(a-b)} \sin (a-b) x+C \]
\[ \int \cos a x \cos b x d x=\frac{1} {2(a+b)} \sin (a+b) x+\frac{1} {2(a-b)} \sin (a-b) x+C \]
\[ \int \frac{d x} {a+b \sin x} =\frac{2} {\sqrt{a^{2} - b ^ {2} } } \arctan ^{n} \frac {\arctan \frac {x} {2} + b} {\sqrt{a^{2} - b^{2} } } + C \]
\[ \int \frac{d x} {a+b \sin x} =\frac{1} {\sqrt{b^{2} -a^{2} } } \ln \left|\frac{\arctan \frac{x} {2} +b-\sqrt{b^{2} -a^{2} } } {\arctan \frac{x} {2} +b+\sqrt{b^{2} -a^{2} } } \right|+C \quad\left(a^{2} <b^{2} \right) \]
\[ \int \frac{d x} {a+b \cos x} =\frac{2} {a+b} \sqrt{\frac{a+b} {a-b} } \arctan \left(\sqrt{\frac{a-b} {a+b} } \tan \frac{x} {2} \right)+C \quad\left(a^{2} >b^{2} \right) \]
\[ \int \frac{d x} {a+b \cos x} =\frac{1} {a+b} \sqrt{\frac{a+b} {b-a} } \ln \left|\frac{\tan \frac{x} {2} +\sqrt{\frac{a+b} {b-a} } } {\tan \frac{x} {2} -\sqrt{\frac{a+b} {b-a} } } \right|+C \quad\left(a^{2} <b^{2} \right) \]
\[ \int \frac{d x} {a^{2} \cos ^{2} x+b^{2} \sin ^{2} x} =\frac{1} {a b} \arctan \left(\frac{b} {a} \tan x\right)+C \]
\[ \int \frac{d x} {a^{2} \cos ^{2} x-b^{2} \sin ^{2} x} =\frac{1} {2 a b} \ln \left|\frac{b \tan x+a} {b \tan x-a} \right|+C \]
\[ \int x \sin a x d x=\frac{1} {a^{2} } \sin a x-\frac{1} {a} x \cos a x+C \]
\[ \int x^{2} \sin a x d x=-\frac{1} {a} x^{2} \cos a x+\frac{2} {a} x d x a x+\frac{2} {a^{2} } x \sin a x+\frac{2} {a^{3} } \cos a x+C \]
\[ \int x \cos a x d x=\frac{1} {a^{2} } \cos a x+\frac{1} {a} x \sin a x+C \]
\[ \int x^{2} \cos a x d x=\frac{1} {a} x^{2} \sin a x+\frac{2} {a^{2} } x \cos ax - \frac {2} {a^{3} } \sin ax + C \]
12. 含有反三角函数的积分 （其中\(a>0\)）
\[ \int \arcsin \frac{x} {a} d x=x \arcsin \frac{x} {a} +\sqrt{a^{2} -x^{2} } +C \]
\[ \int x \arcsin \frac{x} {a} d x=\left(\frac{x^{2} } {2} -\frac{a^{2} } {4} \right) \arcsin \frac{x} {a} +\frac{x} {4} \sqrt{a^{2} -x^{2} } +C \]
\[ \int x^{2} \arcsin \frac{x} {a} d x=\frac{x^{3} } {3} \arcsin \frac{x} {a} +\frac{1} {9} \left(x^{2} +2 a^{2} \right) \sqrt{a^{2} -x^{2} } +C \]
\[ \int \arccos \frac{x} {a} d x=x \arccos \frac{x} {a} -\sqrt{a^{2} -x^{2} } +C \]
\[ \int x \arccos \frac{x} {a} d x=\left(\frac{x^{2} } {2} -\frac{a^{2} } {4} \right) \arccos \frac{x} {4} -\frac{x} {4} \sqrt{a^{2} -x^{2} } +C \]
\[ \int x^{2} \arccos \frac{x} {a} d x=\frac{x^{3} } {a} \arccos \frac{x} {a} -\frac{1} {9} \left(x^{2} +2 a^{2} \right) \sqrt{a^{2} -x^{2} } +C \]
\[ \int \arctan \frac{x} {a} d x=x \arctan \frac{x} {a} -\frac{a} {2} \ln \left(a^{2} +x^{2} \right)+C \]
\[ \int x \arctan \frac{x} {a} d x=\frac{1} {2} \left(a^{2} +x^{2} \right) \arctan \frac{x} {a} -\frac{a} {2} x+C \]
\[ \int x^{2} \arctan \frac{x} {a} d x=\frac{x^{3} } {3} \arctan \frac{x} {a} -\frac{a} {6} x^{2} +\frac{a^{3} } {6} \ln \left(a^{2} +x^{2} \right)+C \]
13. 含有指数函数的积分，其中\(a>0\)
\[ \int a^{x} d x=\frac{1} {\ln a} a^{x} +C \]
\[ \int e^{a x} d x=\frac{1} {a} e^{a x} +C \]
\[ \int x e^{a x} d x=\frac{1} {a^{2} } (a x-1) e^{a x} +C \]
\[ \int x^{n} e^{a x} d x=\frac{1} {a} x^{n} e^{a x} -\frac{n} {a} \int x^{n-1} e^{a x} d x \]
\[ \int x a^{x} d x=\frac{x} {\ln a} a^{x} -\frac{1} {(\ln a)^{2} } a^{x} +C \]
\[ \int x^{n} a^{x} d x=\frac{1} {\ln a} x^{n} a^{x} -\frac{n} {\ln a} \int x^{n-1} a^{x} d x \]
\[ \int e^{a x} \sin b x d x=\frac{1} {a^{2} +b^{2} } e^{a x} (a \sin b x-b \cos b x)+C \]
\[ \int e^{a x} \cos b x d x=\frac{1} {a^{2} +b^{2} } e^{a x} (b \sin b x+a \cos b x)+C \]
\[ \int e^{a x} \sin ^{n} b x d x=\frac{1} {a^{2} +b^{2} n^{2} } e^{a x} \sin ^{n-1} \begin{array} {r} b x(a \sin b x-n b \cos b x) + \frac{n(n-1) b^{2} } {a^{2} +b^{2} n^{2} } \int e^{a x} \sin ^{n-2} b x d x\end{array} \]
\[ \int e^{a x} \cos ^{n} b x d x =\frac{1} {a^{2} +b^{2} n^{2} } e^{a x} \cos ^{n-1} b x(a \cos b x +n b \sin b x) +\frac{n(n-1) b^{2} } {a^{2} +b^{2} n^{2} } \int e^{a x} \cos ^{n-2} b x d x \]
14. 含有对数函数的积分
\[ \int \ln x d x=x \ln x-x+C \]
\[ \int \frac{d x} {x \ln x} =\ln |\ln x|+C \]
\[ \int x^{n} \ln x d x=\frac{1} {n+1} x^{n+1} \left(\ln x-\frac{1} {n+1} \right)+C \]
\[ \int(\ln x)^{n} d x=x(\ln x)^{n} -n \int(\ln x)^{n-1} d x \]
\[ \int x^{m} (\ln x)^{n} d x=\frac{1} {m+1} x^{m+1} (\ln x)^{n} -\frac{n} {m+1} \int x^{m} (\ln x)^{n-1} d x \]
15. 含有双曲函数的积分
\[ \int \sinh x d x=\cosh x+C \]
\[ \int \cosh x d x=\sinh x+C \]
\[ \int \operatorname{th} x d x=\ln \cosh x+C \]
\[ \int \sinh ^{2} x d x=-\frac{x} {2} +\frac{1} {4} \sinh 2 x+C \]
\[ \int \cosh ^{2} x d x=\frac{x} {2} +\frac{1} {4} \sinh 2 x+C \]
16. 定积分
\[ \int_{-\pi} ^{\pi} \cos n x d x=\int_{\pi} ^{\pi} \sin n x d x=0 \]
\[ \int_{-\pi} ^{\pi} \cos m x \cos n x d x=\left \{ \begin{array} {ll} 0, & m \neq n \\ \pi, & m=n\end{array} \right. \]
\[ \int_{-\pi} ^{\pi} \sin m x \sin n x d x=\left \{ \begin{array} {ll} 0, & m \neq n \\ \pi, & m=n\end{array} \right. \]
\[ \int_{0} ^{\pi} \sin m x \sin n x d x=\int_{0} ^{\pi} \cos m x \sin n x d x= \left \{\begin{array} {c} 0, \quad m \neq n \\ \pi / 2, m=n\end{array} \right. \]
\[ I_{n} =\int_{0} ^{\frac{\pi} {2} } \sin ^{n} x d x=\int_{0} ^{\frac{\pi} {2} } \cos ^{n} x d x ; \quad I_{n} =\frac{n-1} {n} I_{n-2} \\ I_{n} =\frac{n-1} {n} \cdot \frac{n-3} {n-2} \ldots \frac{4} {5} \cdot \frac{2} {3} (\text {n为大于1的正奇数} ) ,I_1=1 \\ I_n = \frac{n-1} {n} \cdot \frac{n-3} {n-2} \cdots \frac{3} {4} \cdot \frac{1} {2} \cdot \frac{\pi} {2} (\text {n为正偶数} ) ,I_0= \frac{\pi} {2} \]
17. reference
https://kexue.fm/sci/integral/index.html
