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附录积分表大全（下）

94. $\int{{{\cot }^{n}}ax}dx=\frac{{{\cot }^{n-1}}ax}{a(n-1)}-\int{{{\cot }^{n-2}}ax}dx,n\ne 1$

95. $\int{\sec ax}dx=\frac{1}{a}\ln \left| \sec ax+\tan ax \right|+C$

96. $\int{\csc ax}dx=-\frac{1}{a}\ln \left| \csc ax+\cot ax \right|+C$

97. $\int{{{\sec }^{2}}ax}dx=\frac{1}{a}\tan ax+C$

98. $\int{{{\csc }^{2}}ax}dx=-\frac{1}{a}\cot ax+C$

99. $\int{{{\sec }^{n}}ax}dx=\frac{{{\sec }^{n-2}}ax\tan ax}{a(n-1)}+\frac{n-2}{n-1}\int{{{\sec }^{n-2}}ax}dx,n\ne 1$

100. $\int{{{\csc }^{n}}ax}dx=-\frac{{{\csc }^{n-2}}ax\cot ax}{a(n-1)}+\frac{n-2}{n-1}\int{{{\csc }^{n-2}}ax}dx,n\ne 1$

101. $\int{{{\sec }^{n}}ax\tan ax}dx=\frac{{{\sec }^{n}}ax}{na}+C,n\ne 0$

102. $\int{{{\csc }^{n}}ax\cot ax}dx=-\frac{{{\csc }^{n}}ax}{na}+C,n\ne 0$

103. $\int{{{\sin }^{-1}}ax}dx=x{{\sin }^{-1}}ax+\frac{1}{a}\sqrt{1-{{a}^{2}}{{x}^{2}}}+C$

104. $\int{{{\cos }^{-1}}ax}dx=x{{\cos }^{-1}}ax-\frac{1}{a}\sqrt{1-{{a}^{2}}{{x}^{2}}}+C$

105. $\int{{{\tan }^{-1}}ax}dx=x{{\tan }^{-1}}ax-\frac{1}{2a}\ln \left( 1+{{a}^{2}}{{x}^{2}} \right)+C$

106. $\int{{{x}^{n}}{{\sin }^{-1}}ax}dx=\frac{{{x}^{n+1}}}{n+1}{{\sin }^{-1}}ax-\frac{a}{n+1}\int{\frac{{{x}^{n+1}}dx}{\sqrt{1-{{a}^{2}}{{x}^{2}}}},n\ne -1}$

107. $\int{{{x}^{n}}{{\cos }^{-1}}ax}dx=\frac{{{x}^{n+1}}}{n+1}{{\cos }^{-1}}ax+\frac{a}{n+1}\int{\frac{{{x}^{n+1}}dx}{\sqrt{1-{{a}^{2}}{{x}^{2}}}},n\ne -1}$

108. $\int{{{x}^{n}}{{\tan }^{-1}}ax}dx=\frac{{{x}^{n+1}}}{n+1}{{\tan }^{-1}}ax-\frac{a}{n+1}\int{\frac{{{x}^{n+1}}dx}{1+{{a}^{2}}{{x}^{2}}},n\ne -1}$

109. $\int{{{e}^{ax}}dx}=\frac{1}{a}{{e}^{ax}}+C$

110. $\int{{{b}^{ax}}dx}=\frac{1}{a}\frac{{{b}^{ax}}}{\ln b}+C,(b>0,b\ne 1)$

111. $\int{x{{e}^{ax}}dx}=\frac{{{e}^{ax}}}{{{a}^{2}}}(ax-1)+C$

112. $\int{{{x}^{n}}{{e}^{ax}}dx}=\frac{1}{a}{{x}^{n}}{{e}^{ax}}-\frac{n}{a}\int{{{x}^{n-1}}{{e}^{ax}}}dx$

113. $\int{{{x}^{n}}{{b}^{ax}}dx}=\frac{{{x}^{n}}{{b}^{ax}}}{a\ln b}-\frac{n}{a\ln b}\int{{{x}^{n-1}}{{b}^{ax}}}dx,(b>0,b\ne 1)$

114. $\int{{{e}^{ax}}}\sin bxdx=\frac{{{e}^{ax}}}{{{a}^{2}}+{{b}^{2}}}(a\sin bx-b\cos bx)+C$

115. $\int{{{e}^{ax}}}\cos bxdx=\frac{{{e}^{ax}}}{{{a}^{2}}+{{b}^{2}}}(a\cos bx+b\sin bx)+C$

116. $\int{\ln ax}dx=x\ln ax-x+C$

117. $\int{{{x}^{n}}{{\left( \ln ax \right)}^{m}}}dx=\frac{{{x}^{n+1}}{{\left( \ln ax \right)}^{m}}}{n+1}-\frac{m}{n+1}\int{{{x}^{n}}{{\left( \ln ax \right)}^{m-1}}}dx,n\ne -1$

118. $\int{{{x}^{-1}}{{\left( \ln ax \right)}^{m}}}dx=\frac{{{\left( \ln ax \right)}^{m+1}}}{m+1}+C,m\ne -1$

119. $\int{\frac{dx}{x\ln ax}=\ln \left| \ln ax \right|}+C$

120. $\int{\frac{dx}{\sqrt{2ax-{{x}^{2}}}}}={{\sin }^{-1}}\left( \frac{x-a}{a} \right)+C$

121. $\int{\sqrt{2ax-{{x}^{2}}}}dx=\frac{x-a}{2}\sqrt{2ax-{{x}^{2}}}+\frac{{{a}^{2}}}{2}{{\sin }^{-1}}\left( \frac{x-a}{a} \right)+C$

122. $\int{{{\left( \sqrt{2ax-{{x}^{2}}} \right)}^{n}}}dx=\frac{\left( x-a \right){{\left( \sqrt{2ax-{{x}^{2}}} \right)}^{n}}}{n+1}+\frac{n{{a}^{2}}}{n+1}\int{{{\left( \sqrt{2ax-{{x}^{2}}} \right)}^{n-2}}}dx$

123. $\int{\frac{dx}{{{\left( \sqrt{2ax-{{x}^{2}}} \right)}^{n}}}}=\frac{\left( x-a \right){{\left( \sqrt{2ax-{{x}^{2}}} \right)}^{2-n}}}{\left( n-2 \right){{a}^{2}}}+\frac{n-3}{\left( n-2 \right){{a}^{2}}}\int{\frac{dx}{{{\left( \sqrt{2ax-{{x}^{2}}} \right)}^{n-2}}}}$

124. $\int{x\sqrt{2ax-{{x}^{2}}}}dx=\frac{\left( x+a \right)\left( 2x-3a \right)\sqrt{2ax-{{x}^{2}}}}{6}+\frac{{{a}^{3}}}{2}{{\sin }^{-1}}\left( \frac{x-a}{a} \right)+C$

125. $\int{\frac{\sqrt{2ax-{{x}^{2}}}}{x}}dx=\sqrt{2ax-{{x}^{2}}}+a{{\sin }^{-1}}\left( \frac{x-a}{a} \right)+C$

126. $\int{\frac{\sqrt{2ax-{{x}^{2}}}}{{{x}^{2}}}}dx=-2\sqrt{\frac{2a-x}{x}}-{{\sin }^{-1}}\left( \frac{x-a}{a} \right)+C$

127．$\int{\frac{xdx}{\sqrt{2ax-{{x}^{2}}}}}=a{{\sin }^{-1}}\left( \frac{x-a}{a} \right)-\sqrt{2ax-{{x}^{2}}}+C$

128． $\int{\frac{dx}{x\sqrt{

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附录 积分表大全（下）

附录积分表大全（下）