文章目录
- 含有 ± x − a x + a \sqrt{\pm \frac{x-a}{x+a}} ±x+ax−a 或者 ( x − a ) ( b − x ) \sqrt{(x-a)(b-x)} (x−a)(b−x) 的积分
- 含有三角函数函数的积分
- 含有反三角函数的积分 (其中 a > 0 a>0 a>0)
- 含有指数函数的积分
- 含有对数函数的积分
- 含有双曲函数的积分
- 定积分
含有 ± x − a x + a \sqrt{\pm \frac{x-a}{x+a}} ±x+ax−a 或者 ( x − a ) ( b − x ) \sqrt{(x-a)(b-x)} (x−a)(b−x) 的积分
79. ∫ x − a x − b d x = ( x − b ) x − a x − b + ( b − a ) ln ( ∣ x − a ∣ + ∣ x − b ∣ ) + C \begin{equation} 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 \end{equation} 79.∫x−bx−adx=(x−b)x−bx−a+(b−a)ln(∣x−a∣+∣x−b∣)+C
80. ∫ x − a x − b d x = ( x − b ) x − a x − b + ( b − a ) arcsin x − a b − a + C \begin{equation} 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 \end{equation} 80.∫x−bx−adx=(x−b)x−bx−a+(b−a)arcsinb−ax−a+C
81. ∫ d x ( x − a ) ( x − b ) = 2 arcsin x − a b − a + C ( a < b ) \begin{equation} 81.\,\int\!\! \frac{dx}{\sqrt{(x-a)(x-b)}}=2\arcsin\sqrt{\frac{x-a}{b-a}}+C \quad (a<b) \end{equation} 81.∫(x−a)(x−b)dx=2arcsinb−ax−a+C(a<b)
82. ∫ ( x − a ) ( b − x ) d x = 2 x − a − b 4 ( x − a ) ( b − x ) + ( b − a ) 2 4 arcsin x − a b − a + C ( a < b ) \begin{equation} 82.\,\int\!\! \sqrt{(x-a)(b-x)}\,dx=\frac{2x-a-b}{4}\sqrt{(x-a)(b-x)} +\frac{(b-a)^2}{4}\arcsin\sqrt{\frac{x-a}{b-a}}+C \quad (a<b) \end{equation} 82.∫(x−a)(b−x)dx=42x−a−b(x−a)(b−x)+4(b−a)2arcsinb−ax−a+C(a<b)
含有三角函数函数的积分
83. ∫ sin x d x = − cos x + C \begin{equation} 83.\,\int\!\! \sin x dx = -\cos x +C \end{equation} 83.∫sinxdx=−cosx+C
84. ∫ cos x d x = sin x + C \begin{equation} 84.\,\int\!\! \cos x dx= \sin x +C \end{equation} 84.∫cosxdx=sinx+C
85. ∫ tan x d x − ln ∣ cos x ∣ + C \begin{equation} 85.\,\int\!\! \tan x dx -\ln \vert \cos x \vert +C \end{equation} 85.∫tanxdx−ln∣cosx∣+C
86. ∫ c t g x d x = ln ∣ sin x ∣ + C \begin{equation} 86.\,\int\!\! ctgx dx =\ln \vert \sin x \vert +C \end{equation} 86.∫ctgxdx=ln∣sinx∣+C
87. ∫ sec x d x = ln ∣ tan ( π 4 + x 2 ) ∣ + C = ln ∣ sec x + tan x ∣ + C \begin{equation} 87.\,\int\!\! \sec x dx = \ln \vert \tan (\frac{ \pi}{4} + \frac{x}{2}) \vert + C = \ln \vert \sec x + \tan x \vert +C \end{equation} 87.∫secxdx=ln∣tan(4π+2x)∣+C=ln∣secx+tanx∣+C
88. ∫ csc x d x = ln ∣ tan x 2 ∣ + C = ln ∣ csc x − c t g x ∣ + C \begin{equation} 88.\,\int\!\! \csc x dx= \ln \vert \tan \frac{x}{2} \vert +C = \ln \vert \csc x - ctg x \vert +C \end{equation} 88.∫cscxdx=ln∣tan2x∣+C=ln∣cscx−ctgx∣+C
89. ∫ sec 2 x d x = tan x + C \begin{equation} 89.\,\int\!\! \sec^2 x dx = \tan x +C \end{equation} 89.∫sec2xdx=tanx+C
90. ∫ csc 2 x d x = − c t g x + C \begin{equation} 90.\,\int\!\! \csc ^2 x dx = - ctgx +C \end{equation} 90.∫csc2xdx=−ctgx+C
91. ∫ sec x tan x d x = sec x + C \begin{equation} 91.\,\int\!\! \sec x \tan x dx = \sec x +C \end{equation} 91.∫secxtanxdx=secx+C
92. ∫ csc x d x c t g x d x = − csc x + C \begin{equation} 92.\,\int\!\! \csc x dx ctgx dx = -\csc x +C \end{equation} 92.∫cscxdxctgxdx=−cscx+C
93. ∫ sin 2 x d x = x 2 − 1 4 sin 2 x + C \begin{equation} 93.\,\int\!\! \sin ^2 x dx = \frac{x}{2}- \frac{1}{4}\sin 2x +C \end{equation} 93.∫sin2xdx=2x−41sin2x+C
94. ∫ cos 2 x d x = x 2 + 1 4 sin 2 x + C \begin{equation} 94.\,\int\!\! \cos ^2 x dx = \frac{x}{2} + \frac{1}{4}\sin 2x +C \end{equation} 94.∫cos2xdx=2x+41sin2x+C
95. ∫ sin n x d x = − 1 n sin n − 1 x cos x + n − 1 n ∫ sin n − 2 d x \begin{equation} 95.\,\int\!\! \sin ^n x dx = - \frac{1}{n}\sin ^{n-1}x \cos x + \frac{n-1}{n} \int\!\! \sin ^{n-2}dx \end{equation} 95.∫sinnxdx=−n1sinn−1xcosx+nn−1∫sinn−2dx
96. ∫ cos n x d x = 1 n cos n − 1 x sin x + n − 1 n ∫ cos n − 2 x d x \begin{equation} 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 \end{equation} 96.∫cosnxdx=n1cosn−1xsinx+nn−1∫cosn−2xdx
97. ∫ d x sin n x = − 1 n − 1 . cos x sin n − 1 x + n − 2 n − 1 ∫ d x sin n − 2 x \begin{equation} 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} \end{equation} 97.∫sinnxdx=−n−11.sinn−1xcosx+n−1n−2∫sinn−2xdx
98. ∫ d x cos n x = 1 n − 1 . sin x cos n − 1 x + n − 2 n − 1 ∫ d x cos x n − 2 x \begin{equation} 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} \end{equation} 98.∫cosnxdx=n−11.cosn−1xsinx+n−1n−2∫cosxn−2xdx
99. ∫ cos m sin n x d x = 1 m + n cos m − 1 x sin n + 1 x + m − 1 m + n ∫ cos m − 2 x sin n x d x \begin{equation} 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 \end{equation} 99.∫cosmsinnxdx=m+n1cosm−1xsinn+1x+m+nm−1∫cosm−2xsinnxdx
= − 1 m + 1 cos m + 1 x sin n − 1 x + n − 1 m + n ∫ cos m x sin n − 2 x d x \begin{equation} \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 \notag \end{equation} =−m+11cosm+1xsinn−1x+m+nn−1∫cosmxsinn−2xdx
100. ∫ sin a x cos b x d x = − 1 2 ( a + b ) cos ( a + b ) x − 1 2 ( a − b ) cos ( a − b ) x + C \begin{equation} 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 \end{equation} 100.∫sinaxcosbxdx=−2(a+b)1cos(a+b)x−2(a−b)1cos(a−b)x+C
101. ∫ sin a x sin b x d x = − 1 2 ( a + b ) sin ( a + b ) x 1 2 ( a − b ) sin ( a − b ) x + C \begin{equation} 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 \end{equation} 101.∫sinaxsinbxdx=−2(a+b)1sin(a+b)x2(a−b)1sin(a−b)x+C
102. ∫ cos a x cos b x d x = 1 2 ( a + b ) sin ( a + b ) x + 1 2 ( a − b ) sin ( a − b ) x + C \begin{equation} 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 \end{equation} 102.∫cosaxcosbxdx=2(a+b)1sin(a+b)x+2(a−b)1sin(a−b)x+C
103. ∫ d x a + b sin x = 2 a 2 − b 2 arctan arctan x 2 + b a 2 − b 2 + C ( a 2 > b 2 ) \begin{equation} 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 ) \end{equation} 103.∫a+bsinxdx=a2−b22arctana2−b2arctan2x+b+C(a2>b2)
104. ∫ d x a + b sin x = 1 b 2 − a 2 ln ∣ arctan x 2 + b − b 2 − a 2 arctan x 2 + b + b 2 − a 2 ∣ + C ( a 2 < b 2 ) \begin{equation} 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) \end{equation} 104.∫a+bsinxdx=b2−a21ln arctan2x+b+b2−a2arctan2x+b−b2−a2 +C(a2<b2)
105. ∫ d x a + b cos x = 2 a + b a + b a − b arctan ( a − b a + b tan x 2 ) + C ( a 2 > b 2 ) \begin{equation} 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) \end{equation} 105.∫a+bcosxdx=a+b2a−ba+barctan(a+ba−btan2x)+C(a2>b2)
106. ∫ d x a + b cos x = 1 a + b a + b b − a ln ∣ tan x 2 + a + b b − a tan x 2 − a + b b − a ∣ + C ( a 2 < b 2 ) \begin{equation} 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) \end{equation} 106.∫a+bcosxdx=a+b1b−aa+bln tan2x−b−aa+btan2x+b−aa+b +C(a2<b2)
107. ∫ d x a 2 cos 2 x + b 2 sin 2 x = 1 a b arctan ( b a tan x ) + C \begin{equation} 107.\,\int\!\! \frac{dx}{a^2\cos ^2x + b^2 \sin ^2 x}= \frac{1}{ab} \arctan (\frac{b}{a}\tan x ) +C \end{equation} 107.∫a2cos2x+b2sin2xdx=ab1arctan(abtanx)+C
108. ∫ d x a 2 cos 2 x − b 2 sin 2 x = 1 2 a b ln ∣ b tan x + a b tan x − a ∣ + C \begin{equation} 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 \end{equation} 108.∫a2cos2x−b2sin2xdx=2ab1ln btanx−abtanx+a +C
109. ∫ x sin a x d x = 1 a 2 sin a x − 1 a x cos a x + C \begin{equation} 109.\,\int\!\! x \sin ax dx = \frac{1}{a^2} \sin ax - \frac{1}{a} x \cos ax +C \end{equation} 109.∫xsinaxdx=a21sinax−a1xcosax+C
110. ∫ x 2 sin a x d x = − 1 a x 2 cos a x + 2 a 2 x sin a x + 2 a 3 cos a x + C \begin{equation} 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 \end{equation} 110.∫x2sinaxdx=−a1x2cosax+a22xsinax+a32cosax+C
111. ∫ x cos a x d x = 1 a 2 cos a x + 1 a x sin a x + C \begin{equation} 111.\,\int\!\! x \cos ax dx = \frac{1}{a^2} \cos ax + \frac{1}{a}x \sin ax +C \end{equation} 111.∫xcosaxdx=a21cosax+a1xsinax+C
112. ∫ x 2 cos a x d x = 1 a x 2 sin a x + 2 a 2 x cos a x − 2 a 3 sin a x + C \begin{equation} 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 \end{equation} 112.∫x2cosaxdx=a1x2sinax+a22xcosax−a32sinax+C
含有反三角函数的积分 (其中 a > 0 a>0 a>0)
113. ∫ arcsin x a d x = x arcsin x a + a 2 − x 2 + C \begin{equation} 113.\,\int\!\! \arcsin \frac{x}{a} dx = x \arcsin \frac{x}{a}+ \sqrt{a^2-x^2} +C \end{equation} 113.∫arcsinaxdx=xarcsinax+a2−x2+C
114. ∫ x arcsin x a d x = ( x 2 2 − a 2 4 ) arcsin x a + x 4 a 2 − x 2 + C \begin{equation} 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 \end{equation} 114.∫xarcsinaxdx=(2x2−4a2)arcsinax+4xa2−x2+C
115. ∫ x 2 arcsin x a d x = x 3 3 arcsin x a + 1 9 ( x 2 + 2 a 2 ) a 2 − x 2 + C \begin{equation} 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 \end{equation} 115.∫x2arcsinaxdx=3x3arcsinax+91(x2+2a2)a2−x2+C
116. ∫ arccos x a d x = x arccos x a − a 2 − x 2 + C \begin{equation} 116.\,\int\!\! \arccos \frac{x}{a}dx= x \arccos \frac{x}{a}-\sqrt {a^2-x^2} +C \end{equation} 116.∫arccosaxdx=xarccosax−a2−x2+C
117. ∫ x arccos x a d x = ( x 2 2 − a 2 4 ) arccos x 4 − x 4 a 2 − x 2 + C \begin{equation} 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 \end{equation} 117.∫xarccosaxdx=(2x2−4a2)arccos4x−4xa2−x2+C
118. ∫ x 2 arccos x a d x = x 3 a arccos x a − 1 9 ( x 2 + 2 a 2 ) a 2 − x 2 + C \begin{equation} 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 \end{equation} 118.∫x2arccosaxdx=ax3arccosax−91(x2+2a2)a2−x2+C
119. ∫ arctan x a d x = x arctan x a − a 2 ln ( a 2 + x 2 ) + C \begin{equation} 119.\,\int\!\! \arctan \frac{x}{a} dx =x \arctan \frac{x}{a} - \frac{a}{2}\ln(a^2+x^2)+C \end{equation} 119.∫arctanaxdx=xarctanax−2aln(a2+x2)+C
120. ∫ x arctan x a d x = 1 2 ( a 2 + x 2 ) arctan x a − a 2 x + C \begin{equation} 120.\,\int\!\! x\arctan \frac{x}{a}dx = \frac{1}{2}(a^2+x^2)\arctan \frac{x}{a}-\frac{a}{2}x+C \end{equation} 120.∫xarctanaxdx=21(a2+x2)arctanax−2ax+C
121. ∫ x 2 arctan x a d x = x 3 3 arctan x a − a 6 x 2 + a 3 6 ln ( a 2 + x 2 ) + C \begin{equation} 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 \end{equation} 121.∫x2arctanaxdx=3x3arctanax−6ax2+6a3ln(a2+x2)+C
含有指数函数的积分
122. ∫ a x d x = 1 ln a a x + C \begin{equation} 122.\,\int\!\! a^x dx = \frac{1}{\ln a} a^x +C \end{equation} 122.∫axdx=lna1ax+C
123. ∫ e a x d x = 1 a e a x + C \begin{equation} 123.\,\int\!\! e^{ax}dx = \frac{1}{a} e ^{ax}+C \end{equation} 123.∫eaxdx=a1eax+C
124. ∫ x e a x d x = 1 a 2 ( a x − 1 ) e a x + c \begin{equation} 124.\,\int\!\! xe^{ax}dx=\frac{1}{a^2}(ax-1)e^{ax}+c \end{equation} 124.∫xeaxdx=a21(ax−1)eax+c
125. ∫ x n e a x d x = 1 a x n e a x − n a ∫ x n − 1 e a x d x \begin{equation} 125.\,\int\!\! x^n e^{ax}dx=\frac{1}{a}x^n e^{ax}- \frac{n}{a}\int\!\! x^{n-1} e^{ax}dx \end{equation} 125.∫xneaxdx=a1xneax−an∫xn−1eaxdx
126. ∫ x a x d x = x ln a a x − 1 ( ln a ) 2 a x + C \begin{equation} 126.\,\int\!\! xa^x dx= \frac{x}{\ln a}a^x - \frac{1}{(\ln a)^2}a^x+C \end{equation} 126.∫xaxdx=lnaxax−(lna)21ax+C
127. ∫ x n a x d x = 1 ln a x n a x − n ln a ∫ x n − 1 a x d x \begin{equation} 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 \end{equation} 127.∫xnaxdx=lna1xnax−lnan∫xn−1axdx
128. ∫ e a x sin b x d x = 1 a 2 + b 2 e a x ( a sin b x − b cos b x ) + C \begin{equation} 128.\,\int\!\! e^{ax} \sin bx dx = \frac{1}{a^2+b^2}e^{ax}(a\sin bx -b \cos bx)+C \end{equation} 128.∫eaxsinbxdx=a2+b21eax(asinbx−bcosbx)+C
129. ∫ e a x cos b x d x = 1 a 2 + b 2 e a x ( b sin b x + a cos b x ) + C \begin{equation} 129.\,\int\!\! e^{ax}\cos bx dx = \frac{1}{a^2+b^2} e^{ax} (b \sin bx + a \cos bx) +C \end{equation} 129.∫eaxcosbxdx=a2+b21eax(bsinbx+acosbx)+C
130. ∫ e a x sin n b x d x = 1 a 2 + b 2 n 2 e a x sin n − 1 b x ( a sin b x − n b cos b x ) \begin{equation} 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) \end{equation} 130.∫eaxsinnbxdx=a2+b2n21eaxsinn−1bx(asinbx−nbcosbx)
+ n ( n − 1 ) b 2 a 2 + b 2 n 2 ∫ e a x sin n − 2 b x d x \begin{equation} \qquad \qquad \qquad \qquad \qquad+ \frac{n(n-1)b^2}{a^2+b^2n^2}\int\!\! e^{ax}\sin ^{n-2}bx dx \notag \end{equation} +a2+b2n2n(n−1)b2∫eaxsinn−2bxdx
131. ∫ e a x cos n b x d x = 1 a 2 + b 2 n 2 e a x cos n − 1 b x ( a cos b x + n b sin b x ) \begin{equation} 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) \end{equation} 131.∫eaxcosnbxdx=a2+b2n21eaxcosn−1bx(acosbx+nbsinbx)
+ n ( n − 1 ) b 2 a 2 + b 2 n 2 ∫ e a x cos n − 2 b x d x \begin{equation} \qquad \qquad \qquad \qquad \qquad + \frac{n(n-1)b^2}{a^2+b^2n^2} \int\!\! e^{ax} \cos ^{n-2}bx dx \notag \end{equation} +a2+b2n2n(n−1)b2∫eaxcosn−2bxdx
含有对数函数的积分
132. ∫ ln x d x = x ln x − x + C \begin{equation} 132. \, \int\!\!\ln x dx=x\ln x -x +C \end{equation} 132.∫lnxdx=xlnx−x+C
133. ∫ d x x ln x = ln ∣ ln x ∣ + C \begin{equation} 133.\,\int\!\! \frac {dx}{x\ln x}= \ln \vert \ln x \vert +C \end{equation} 133.∫xlnxdx=ln∣lnx∣+C
134. ∫ x n ln x d x = 1 n + 1 x n + 1 ( ln x − 1 n + 1 ) + C \begin{equation} 134. \,\int\!\! x^n\ln x dx=\frac{1}{n+1}x^{n+1}(\ln x - \frac{1}{n+1})+C \end{equation} 134.∫xnlnxdx=n+11xn+1(lnx−n+11)+C
135. ∫ ( ln x ) n d x = x ( ln x ) n − n ∫ ( ln x ) n − 1 d x \begin{equation} 135. \,\int\!\! (\ln x)^n dx=x(\ln x)^n - n \int\!\!(\ln x)^{n-1} dx \end{equation} 135.∫(lnx)ndx=x(lnx)n−n∫(lnx)n−1dx
136. ∫ x m ( ln x ) n d x = 1 m + 1 x m + 1 ( ln x ) n − n m + 1 ∫ x m ( ln x ) n − 1 d x \begin{equation} 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 \end{equation} 136.∫xm(lnx)ndx=m+11xm+1(lnx)n−m+1n∫xm(lnx)n−1dx
含有双曲函数的积分
137. ∫ sinh x d x = cosh x + C \begin{equation} 137. \,\int\!\! \sinh x dx = \cosh x +C \end{equation} 137.∫sinhxdx=coshx+C
138. ∫ cosh x d x = sinh x + C \begin{equation} 138. \,\int\!\! \cosh x dx = \sinh x +C \end{equation} 138.∫coshxdx=sinhx+C
139. ∫ t h x d x = ln cosh x + C \begin{equation} 139. \,\int\!\! th x dx =\ln \cosh x +C \end{equation} 139.∫thxdx=lncoshx+C
140. ∫ sinh 2 x d x = − x 2 + 1 4 sinh 2 x + C \begin{equation} 140. \,\int\!\! \sinh ^2 x dx = -\frac{x}{2} + \frac{1}{4} \sinh 2x +C \end{equation} 140.∫sinh2xdx=−2x+41sinh2x+C
141. ∫ cosh 2 x d x = x 2 + 1 4 sinh 2 x + C \begin{equation} 141. \,\int\!\! \cosh^2 x dx = \frac{x}{2}+\frac{1}{4} \sinh 2x +C \end{equation} 141.∫cosh2xdx=2x+41sinh2x+C
定积分
142. ∫ − π π cos n x d x = ∫ π π sin n x d x = 0 \begin{equation} 142. \int^{\pi}_{-\pi} \cos nx dx = \int^{\pi}_{\pi}\sin nx dx=0 \end{equation} 142.∫−ππcosnxdx=∫ππsinnxdx=0
143. ∫ − π π cos m x sin n x d x = 0 \begin{equation} 143. \,\int^ {\pi} _{-\pi} \cos mx \sin nx dx =0 \end{equation} 143.∫−ππcosmxsinnxdx=0
144. ∫ − π π cos m x cos n x d x = { 0 , m ≠ n π , m = n \begin{equation} 144. \,\int^{\pi} _{-\pi} \cos mx \cos nx dx =\left \{ \begin{array}{cc} 0, & m \neq n \\ \pi, & m=n \end{array} \right. \end{equation} 144.∫−ππcosmxcosnxdx={0,π,m=nm=n
145. ∫ − π π sin m x sin n x d x = { 0 , m ≠ n π , m = n \begin{equation} 145. \,\int ^\pi _{-\pi} \sin mx \sin nx dx = \left \{ \begin{array}{cc} 0, & m \neq n \\ \pi, & m=n \end{array} \right. \end{equation} 145.∫−ππsinmxsinnxdx={0,π,m=nm=n
146. ∫ 0 π sin m x sin n x d x = ∫ 0 π cos m x cos n x d x = { 0 , m ≠ n π / 2 , m = n \begin{equation} 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. \end{equation} 146.∫0πsinmxsinnxdx=∫0πcosmxcosnxdx={0,π/2,m=nm=n
147. I n = ∫ 0 π 2 sin n x d x = ∫ 0 π 2 cos n x d x \begin{equation} 147. I_n =\,\int ^{\frac{\pi}{2}} _0 \sin ^n x dx = \int ^{\frac{\pi}{2}} _0 \cos ^n x dx \end{equation} 147.In=∫02πsinnxdx=∫02πcosnxdx
I n = n − 1 n I n − 2 \begin{equation} \qquad I_n= \frac{n-1}{n} I _{n-2} \notag \end{equation} In=nn−1In−2
{ I n = n − 1 n . n − 3 n − 2 . … 4 5 . 2 3 (n为大于1的正奇数) , I 1 = 1 I n = n − 1 n . n − 3 n − 2 . … 3 4 . 1 2 . π 2 (n为正偶数) , I 0 = π 2 \begin{equation} \qquad \left \{ \begin{aligned} I_n&= \frac{n-1}{n}.\frac{n-3}{n-2}.\dots \frac{4}{5} .\frac{2}{3} \quad \text{(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} \quad \text{(n为正偶数)},I_0= \frac{\pi}{2} \end{aligned} \right.\notag \end{equation} ⎩ ⎨ ⎧InIn=nn−1.n−2n−3.…54.32(n为大于1的正奇数),I1=1=nn−1.n−2n−3.…43.21.2π(n为正偶数),I0=2π
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