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

解答

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

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

求解步骤

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

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

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

乘开

\frac{x + 1 - x - 1}{( x + 1 ) ^{2} - ( x - 1 ) ^{2}} : \frac{x + 1}{( x + 1 ) ^{2} - ( x - 1 ) ^{2}} - \frac{x - 1}{( x + 1 ) ^{2} - ( x - 1 ) ^{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 + 1}{( x + 1 ) ^{2} - ( x - 1 ) ^{2}} d x - \int \frac{x - 1}{( x + 1 ) ^{2} - ( x - 1 ) ^{2}} d x

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

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

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

解答补常数

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

a re a f (x) = x^{2} - 1, g (x) = - x + 2, 0, 1

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

\int_{0}^{3} \frac{1}{9 - x ^{2}} d x

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

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