integral of 1/(sqrt(x+3)-\sqrt{x+2)}

解答

\int \frac{1}{x + 3 - x + 2} d x

\frac{2}{3} (x + 3)^{\frac{3}{2}} + \frac{2}{3} (x + 2)^{\frac{3}{2}} + C

求解步骤

\int \frac{1}{x + 3 - x + 2} d x

\frac{1}{x + 3 - x + 2} = \frac{x + 3 + x + 2}{( x + 3 ) ^{2} - ( x + 2 ) ^{2}}

= \int \frac{x + 3 + x + 2}{( x + 3 ) ^{2} - ( x + 2 ) ^{2}} d x

乘开

\frac{x + 3 + x + 2}{( x + 3 ) ^{2} - ( x + 2 ) ^{2}} : \frac{x + 3}{( x + 3 ) ^{2} - ( x + 2 ) ^{2}} + \frac{x + 2}{( x + 3 ) ^{2} - ( x + 2 ) ^{2}}

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x + 3}{( x + 3 ) ^{2} - ( x + 2 ) ^{2}} d x + \int \frac{x + 2}{( x + 3 ) ^{2} - ( x + 2 ) ^{2}} d x

\int \frac{x + 3}{( x + 3 ) ^{2} - ( x + 2 ) ^{2}} d x = \frac{2}{3} (x + 3)^{\frac{3}{2}}

\int \frac{x + 2}{( x + 3 ) ^{2} - ( x + 2 ) ^{2}} d x = \frac{2}{3} (x + 2)^{\frac{3}{2}}

= \frac{2}{3} (x + 3)^{\frac{3}{2}} + \frac{2}{3} (x + 2)^{\frac{3}{2}}

解答补常数

= \frac{2}{3} (x + 3)^{\frac{3}{2}} + \frac{2}{3} (x + 2)^{\frac{3}{2}} + C

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