integral of ((x+3))/(sqrt(1-x^2))

解答

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

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

求解步骤

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

乘开

\frac{x + 3}{1 - x ^{2}} : \frac{x}{1 - x ^{2}} + \frac{3}{1 - x ^{2}}

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

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

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

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

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

解答补常数

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

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

\int \frac{sin ( 4 x )}{tan ( 4 x )} d x

\int 9 x^{2} (3 x^{3} - 4)^{5} d x

\int \frac{1}{( 7 - t ^{2} ) ^{\frac{3}{2}}} d t

\int \frac{x ^{2}}{( 9 - x ^{2} ) ^{3}} d x