换元法

例1<span class="formula-wrapper" contenteditable="false" data-tex="\int\sqrt{a^2-x^2}dx">\int\sqrt{a^2-x^2}dx

例2\int\frac{1}{\sqrt{x^2+a^2}}dx

例3\int\frac{1}{\sqrt{x^2-a^2}}dx

1、x=a\sin{t} 或x=a\cos{t} ，适用于被积函数含有\sqrt{a^2-x^2}

考虑例1<span class="formula-wrapper" contenteditable="false" data-tex="\int\sqrt{a^2-x^2}dx">\int\sqrt{a^2-x^2}dx ，\int\sqrt{a^2-x^2}dx=\int\sqrt{a^2-a^2\sin^2{t}}{\cdot}a\cos{t}dt =\int{a^2}\cos^2{t}dt =\int\frac{1}{2}+\frac{1}{2}\cos{2t}dt =\frac{t}{2}+\frac{1}{4}\sin{2t}+C

=\frac{1}{2}\arcsin{\frac{x}{a}}+\frac{1}{2}\frac{x}{a}\sqrt{1-\frac{x^2}{a^2}}+C =\frac{a^2}{2}\arcsin{\frac{x}{a}}+\frac{1}{2}x\sqrt{a^2-x^2}+C

2、x=a\tan{t} ，适用于被积函数含有\sqrt{a^2+x^2}

考虑例2\int\frac{1}{\sqrt{x^2+a^2}}dx ，\int\frac{1}{\sqrt{x^2+a^2}}dx=\int\frac{1}{\sqrt{a^2\tan^2{t}+a^2}}\frac{a}{\cos^2{t}}dt =\int\sec{t}dt

=\int\sec{t}\frac{\sec{t}+\tan{t}}{\sec{t}+\tan{t}}dt =\int\frac{\sec^2{t}+\frac{\sin{t}}{\cos^2{t}}}{\sec{t}+\tan{t}}dt =\ln|\sec{t}+\tan{t}|+C

=\ln(\sqrt{1+\frac{x^2}{a^2}}+\frac{x}{a})+C =\ln(\sqrt{a^2+x^2}+x)+C

3、x=a\sec{t} ，适用于被积函数含有\sqrt{x^2-a^2}

考虑例3\int\frac{1}{\sqrt{x^2-a^2}}dx ，\int\frac{1}{\sqrt{x^2-a^2}}dx=\int\frac{a\sin{t}\sec^2{t}}{a\tan{x}}dt=\int\sec{t}dt

=\int\sec{t}\frac{\sec{t}+\tan{t}}{\sec{t}+\tan{t}}dt =\int\frac{\sec^2{t}+\frac{\sin{t}}{\cos^2{t}}}{\sec{t}+\tan{t}}dt =\ln|\sec{t}+\tan{t}|+C

\int\ln(\frac{x}{a}+\sqrt{\frac{x^2}{a^2}-1})dx+C =\int\ln(x+\sqrt{x^2-a^2})dx+C

双曲正弦sh{x}=\sinh{x}=\frac{e^x-e^{-x}}{2}

双曲余弦chx=\cosh{x}=\frac{e^x+e^{-x}}{2}

ch^2x-sh^2x=1

1、x=asht ，适用于被积函数含有\sqrt{x^2+a^2}

考虑例2\int\frac{1}{\sqrt{x^2+a^2}}dx ，\int\frac{1}{\sqrt{x^2+a^2}}dx=\int\frac{a\cosh{t}}{\sqrt{a^2\sinh^2{t}+a^2}}dt =\int1dt=t+C

=\ln(\cosh{t}+\sinh{t})+C =\ln(\sqrt{1+\frac{x^2}{a^2}}+\frac{x}{a})+C =\ln(\sqrt{x^2+a^2}+x)+C

2、x=acht ，适用于被积函数含有\sqrt{x^2-a^2}

考虑例3\int\frac{1}{\sqrt{x^2-a^2}}dx ，\int\frac{1}{\sqrt{x^2-a^2}}dx=\int\frac{a\sinh{t}}{\sqrt{a^2\cosh{t}-a^2}}dt=\int1dt=t+C

=\ln(\cosh{t}+\sinh{t})+C =\ln(\frac{x}{a}+\sqrt{\frac{x^2}{a^2}-1})+C =\ln(x+\sqrt{x^2-a^2})+C