integral of x(1-x^4)^{3/2}

解答

\int x (1 - x^{4})^{\frac{3}{2}} d x

\frac{1}{2} (x^{2} (1 - x^{4})^{\frac{3}{2}} + \frac{3}{8} (arcsin (x^{2}) - \frac{1}{4} sin (4 arcsin (x^{2})))) + C

求解步骤

\int x (1 - x^{4})^{\frac{3}{2}} d x

使用换元积分法

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

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int (1 - u^{2})^{\frac{3}{2}} d u

使用分布积分法

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

\int - 3 u^{2} (1 - u^{2})^{\frac{1}{2}} d u = - \frac{3}{8} (arcsin (u) - \frac{1}{4} sin (4 arcsin (u)))

= \frac{1}{2} (u (1 - u^{2})^{\frac{3}{2}} - (- \frac{3}{8} (arcsin (u) - \frac{1}{4} sin (4 arcsin (u)))))

u = x^{2}

代回

= \frac{1}{2} (x^{2} (1 - (x^{2})^{2})^{\frac{3}{2}} - (- \frac{3}{8} (arcsin (x^{2}) - \frac{1}{4} sin (4 arcsin (x^{2})))))

化简

\frac{1}{2} (x^{2} (1 - (x^{2})^{2})^{\frac{3}{2}} - (- \frac{3}{8} (arcsin (x^{2}) - \frac{1}{4} sin (4 arcsin (x^{2}))))) : \frac{1}{2} (x^{2} (1 - x^{4})^{\frac{3}{2}} + \frac{3}{8} (arcsin (x^{2}) - \frac{1}{4} sin (4 arcsin (x^{2}))))

= \frac{1}{2} (x^{2} (1 - x^{4})^{\frac{3}{2}} + \frac{3}{8} (arcsin (x^{2}) - \frac{1}{4} sin (4 arcsin (x^{2}))))

解答补常数

= \frac{1}{2} (x^{2} (1 - x^{4})^{\frac{3}{2}} + \frac{3}{8} (arcsin (x^{2}) - \frac{1}{4} sin (4 arcsin (x^{2})))) + C

\int (e^{3 x} + 6)^{3} e^{2 x} d x

\int \frac{( x + 1 )}{4 4} (x - 1) d x

\int \frac{11 x ^{2}}{( 100 + x ^{2} ) ^{2}} d x

\int cos (\frac{x}{a}) d x

\int \frac{x cos ^{2} ( x ^{2} )}{sin ( x ^{2} )} d x