丰县数据网站建设多少钱,做物流的网站,中国建筑信息资讯网,最火的做牛排沙拉网站文章目录 含有 x − a x a \sqrt{\pm \frac{x-a}{xa}} xax−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 a0 a0)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… 文章目录 含有 ± x − a x a \sqrt{\pm \frac{x-a}{xa}} ±xax−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 a0 a0)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}{xa}} ±xax−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−a dx(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−a dx(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 (ab) \end{equation} 81.∫(x−a)(x−b) dx2arcsinb−ax−a C(ab) 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 (ab) \end{equation} 82.∫(x−a)(b−x) dx42x−a−b(x−a)(b−x) 4(b−a)2arcsinb−ax−a C(ab)
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−cosxC 84. ∫ cos x d x sin x C \begin{equation} 84.\,\int\!\! \cos x dx \sin x C \end{equation} 84.∫cosxdxsinxC 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.∫ctgxdxln∣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.∫secxdxln∣tan(4π2x)∣Cln∣secxtanx∣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.∫cscxdxln∣tan2x∣Cln∣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.∫sec2xdxtanxC 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−ctgxC 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.∫secxtanxdxsecxC 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−cscxC 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.∫sin2xdx2x−41sin2xC 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.∫cos2xdx2x41sin2xC 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−1xcosxnn−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.∫cosnxdxn1cosn−1xsinxnn−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−1xcosxn−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.∫cosnxdxn−11.cosn−1xsinxn−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}{mn}\cos^{m-1}x \sin ^{n1}x \frac{m-1}{mn} \int\!\! \cos ^ {m-2} x \sin ^n x dx \end{equation} 99.∫cosmsinnxdxmn1cosm−1xsinn1xmnm−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}{m1}\cos ^{m1}x \sin ^{n-1}x \frac{n-1}{mn} \int\!\! \cos ^m x \sin ^{n-2} x dx \notag \end{equation} −m11cosm1xsinn−1xmnn−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(ab)}\cos (ab)x - \frac{1}{2(a-b)} \cos (a-b)x C \end{equation} 100.∫sinaxcosbxdx−2(ab)1cos(ab)x−2(a−b)1cos(a−b)xC 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(ab)} \sin (ab) x\frac{1}{2(a-b)} \sin (a-b)x C \end{equation} 101.∫sinaxsinbxdx−2(ab)1sin(ab)x2(a−b)1sin(a−b)xC 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(ab)} \sin (ab)x \frac{1}{2(a-b)} \sin (a-b)x C \end{equation} 102.∫cosaxcosbxdx2(ab)1sin(ab)x2(a−b)1sin(a−b)xC 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}{ab\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.∫absinxdxa2−b2 2arctana2−b2 arctan2xbC(a2b2) 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}{ab \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^2b^2) \end{equation} 104.∫absinxdxb2−a2 1ln arctan2xbb2−a2 arctan2xb−b2−a2 C(a2b2) 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}{ab \cos x} \frac{2}{ab} \sqrt{ \frac{ab}{a-b}} \arctan \Bigg ( \sqrt { \frac{a-b}{ab}} \tan \frac{x}{2} \Bigg ) C \qquad (a^2b^2) \end{equation} 105.∫abcosxdxab2a−bab arctan(aba−b tan2x)C(a2b2) 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}{ab \cos x} \frac{1}{ab}\sqrt{ \frac{ab}{b-a}} \ln \Bigg \vert \frac{\tan \frac{x}{2} \sqrt {\frac{ab}{b-a}}}{\tan \frac{x}{2}- \sqrt{\frac{ab}{b-a}}} \Bigg \vert C \qquad (a^2b^2) \end{equation} 106.∫abcosxdxab1b−aab ln tan2x−b−aab tan2xb−aab C(a2b2) 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.∫a2cos2xb2sin2xdxab1arctan(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−b2sin2xdx2ab1ln btanx−abtanxa 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.∫xsinaxdxa21sinax−a1xcosaxC 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−a1x2cosaxa22xsinaxa32cosaxC 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.∫xcosaxdxa21cosaxa1xsinaxC 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.∫x2cosaxdxa1x2sinaxa22xcosax−a32sinaxC
含有反三角函数的积分 (其中 a 0 a0 a0) 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.∫arcsinaxdxxarcsinaxa2−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)arcsinax4xa2−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^22a^2)\sqrt{a^2-x^2}C \end{equation} 115.∫x2arcsinaxdx3x3arcsinax91(x22a2)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.∫arccosaxdxxarccosax−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.∫x2arccosaxdxax3arccosax−91(x22a2)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^2x^2)C \end{equation} 119.∫arctanaxdxxarctanax−2aln(a2x2)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^2x^2)\arctan \frac{x}{a}-\frac{a}{2}xC \end{equation} 120.∫xarctanaxdx21(a2x2)arctanax−2axC 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^2x^2)C \end{equation} 121.∫x2arctanaxdx3x3arctanax−6ax26a3ln(a2x2)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.∫axdxlna1axC 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.∫eaxdxa1eaxC 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.∫xeaxdxa21(ax−1)eaxc 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.∫xneaxdxa1xneax−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^xC \end{equation} 126.∫xaxdxlnaxax−(lna)21axC 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.∫xnaxdxlna1xnax−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^2b^2}e^{ax}(a\sin bx -b \cos bx)C \end{equation} 128.∫eaxsinbxdxa2b21eax(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^2b^2} e^{ax} (b \sin bx a \cos bx) C \end{equation} 129.∫eaxcosbxdxa2b21eax(bsinbxacosbx)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^2b^2n^2}e^{ax}\sin ^{n-1}bx (a \sin bx - nb \cos bx) \end{equation} 130.∫eaxsinnbxdxa2b2n21eaxsinn−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^2b^2n^2}\int\!\! e^{ax}\sin ^{n-2}bx dx \notag \end{equation} a2b2n2n(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^2b^2n^2} e^{ax} \cos ^{n-1} bx (a\cos bx nb \sin bx) \end{equation} 131.∫eaxcosnbxdxa2b2n21eaxcosn−1bx(acosbxnbsinbx) 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^2b^2n^2} \int\!\! e^{ax} \cos ^{n-2}bx dx \notag \end{equation} a2b2n2n(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 dxx\ln x -x C \end{equation} 132.∫lnxdxxlnx−xC 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.∫xlnxdxln∣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}{n1}x^{n1}(\ln x - \frac{1}{n1})C \end{equation} 134.∫xnlnxdxn11xn1(lnx−n11)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 dxx(\ln x)^n - n \int\!\!(\ln x)^{n-1} dx \end{equation} 135.∫(lnx)ndxx(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}{m1}x^{m1}(\ln x)^n - \frac {n}{m1} \int\!\! x^m (\ln x) ^{n-1} dx \end{equation} 136.∫xm(lnx)ndxm11xm1(lnx)n−m1n∫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.∫sinhxdxcoshxC 138. ∫ cosh x d x sinh x C \begin{equation} 138. \,\int\!\! \cosh x dx \sinh x C \end{equation} 138.∫coshxdxsinhxC 139. ∫ t h x d x ln cosh x C \begin{equation} 139. \,\int\!\! th x dx \ln \cosh x C \end{equation} 139.∫thxdxlncoshxC 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−2x41sinh2xC 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.∫cosh2xdx2x41sinh2xC
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 dx0 \end{equation} 142.∫−ππcosnxdx∫ππsinnxdx0 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.∫−ππcosmxsinnxdx0 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, mn \end{array} \right. \end{equation} 144.∫−ππcosmxcosnxdx{0,π,mnmn 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, mn \end{array} \right. \end{equation} 145.∫−ππsinmxsinnxdx{0,π,mnmn 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, mn \end{array} \right. \end{equation} 146.∫0πsinmxsinnxdx∫0πcosmxcosnxdx{0,π/2,mnmn 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} Innn−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_11 \\ 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} ⎩ ⎨ ⎧InInnn−1.n−2n−3.…54.32(n为大于1的正奇数),I11nn−1.n−2n−3.…43.21.2π(n为正偶数),I02π 文章转载自： http://www.morning.tznlz.cn.gov.cn.tznlz.cn http://www.morning.dxzcr.cn.gov.cn.dxzcr.cn http://www.morning.bqpg.cn.gov.cn.bqpg.cn http://www.morning.ygqjn.cn.gov.cn.ygqjn.cn http://www.morning.fdrch.cn.gov.cn.fdrch.cn http://www.morning.yqyhr.cn.gov.cn.yqyhr.cn http://www.morning.pbmg.cn.gov.cn.pbmg.cn http://www.morning.qkdjq.cn.gov.cn.qkdjq.cn http://www.morning.pxwjp.cn.gov.cn.pxwjp.cn http://www.morning.wdpbq.cn.gov.cn.wdpbq.cn http://www.morning.slysg.cn.gov.cn.slysg.cn http://www.morning.xqcgb.cn.gov.cn.xqcgb.cn http://www.morning.mcqhb.cn.gov.cn.mcqhb.cn http://www.morning.gtcym.cn.gov.cn.gtcym.cn http://www.morning.qwfl.cn.gov.cn.qwfl.cn http://www.morning.bnxfj.cn.gov.cn.bnxfj.cn http://www.morning.qxmnf.cn.gov.cn.qxmnf.cn http://www.morning.yhjlg.cn.gov.cn.yhjlg.cn http://www.morning.clbsd.cn.gov.cn.clbsd.cn http://www.morning.xkwrb.cn.gov.cn.xkwrb.cn http://www.morning.rghkg.cn.gov.cn.rghkg.cn http://www.morning.lqynj.cn.gov.cn.lqynj.cn http://www.morning.ncqzb.cn.gov.cn.ncqzb.cn http://www.morning.pqcrz.cn.gov.cn.pqcrz.cn http://www.morning.wbysj.cn.gov.cn.wbysj.cn http://www.morning.svtxeu.com.gov.cn.svtxeu.com http://www.morning.fdfdz.cn.gov.cn.fdfdz.cn http://www.morning.qnbgh.cn.gov.cn.qnbgh.cn http://www.morning.rhfh.cn.gov.cn.rhfh.cn http://www.morning.dnbhd.cn.gov.cn.dnbhd.cn http://www.morning.fbxlj.cn.gov.cn.fbxlj.cn http://www.morning.zrdhd.cn.gov.cn.zrdhd.cn http://www.morning.rytps.cn.gov.cn.rytps.cn http://www.morning.nzzws.cn.gov.cn.nzzws.cn http://www.morning.sfzwm.cn.gov.cn.sfzwm.cn http://www.morning.fnxzk.cn.gov.cn.fnxzk.cn http://www.morning.3dcb8231.cn.gov.cn.3dcb8231.cn http://www.morning.mzhh.cn.gov.cn.mzhh.cn http://www.morning.ynjhk.cn.gov.cn.ynjhk.cn http://www.morning.pcngq.cn.gov.cn.pcngq.cn http://www.morning.flfxb.cn.gov.cn.flfxb.cn http://www.morning.wphfl.cn.gov.cn.wphfl.cn http://www.morning.tzjqm.cn.gov.cn.tzjqm.cn http://www.morning.bfmrq.cn.gov.cn.bfmrq.cn http://www.morning.wkkqw.cn.gov.cn.wkkqw.cn http://www.morning.ljxxl.cn.gov.cn.ljxxl.cn http://www.morning.sqxr.cn.gov.cn.sqxr.cn http://www.morning.ltffk.cn.gov.cn.ltffk.cn http://www.morning.pprxs.cn.gov.cn.pprxs.cn http://www.morning.mksny.cn.gov.cn.mksny.cn http://www.morning.ymhzd.cn.gov.cn.ymhzd.cn http://www.morning.jbgzy.cn.gov.cn.jbgzy.cn http://www.morning.lhztj.cn.gov.cn.lhztj.cn http://www.morning.jrhmh.cn.gov.cn.jrhmh.cn http://www.morning.qkkmd.cn.gov.cn.qkkmd.cn http://www.morning.hwnqg.cn.gov.cn.hwnqg.cn http://www.morning.lmmh.cn.gov.cn.lmmh.cn http://www.morning.fxzlg.cn.gov.cn.fxzlg.cn http://www.morning.xdxpq.cn.gov.cn.xdxpq.cn http://www.morning.nlkjq.cn.gov.cn.nlkjq.cn http://www.morning.xqwq.cn.gov.cn.xqwq.cn http://www.morning.mjglk.cn.gov.cn.mjglk.cn http://www.morning.xdlwm.cn.gov.cn.xdlwm.cn http://www.morning.skbhl.cn.gov.cn.skbhl.cn http://www.morning.mlcnh.cn.gov.cn.mlcnh.cn http://www.morning.rkwlg.cn.gov.cn.rkwlg.cn http://www.morning.bcngs.cn.gov.cn.bcngs.cn http://www.morning.yzxhk.cn.gov.cn.yzxhk.cn http://www.morning.brkc.cn.gov.cn.brkc.cn http://www.morning.rui931.cn.gov.cn.rui931.cn http://www.morning.txlxr.cn.gov.cn.txlxr.cn http://www.morning.ksqyj.cn.gov.cn.ksqyj.cn http://www.morning.jzccn.cn.gov.cn.jzccn.cn http://www.morning.cqwb25.cn.gov.cn.cqwb25.cn http://www.morning.xrwbc.cn.gov.cn.xrwbc.cn http://www.morning.qbksx.cn.gov.cn.qbksx.cn http://www.morning.grxyx.cn.gov.cn.grxyx.cn http://www.morning.dmzmy.cn.gov.cn.dmzmy.cn http://www.morning.sbrrf.cn.gov.cn.sbrrf.cn http://www.morning.sxbgc.cn.gov.cn.sxbgc.cn
