integral of tan^3(x)sec^{3/2}(x)

解答

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

\frac{2}{7} sec^{\frac{7}{2}} (x) - \frac{2}{3} sec^{\frac{3}{2}} (x) + C

求解步骤

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

使用三角恒等式改写

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

使用换元积分法

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

乘开

u^{\frac{1}{2}} (u^{2} - 1) : u^{\frac{5}{2}} - u^{\frac{1}{2}}

= \int u^{\frac{5}{2}} - u^{\frac{1}{2}} d u

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 u^{\frac{5}{2}} d u - \int u^{\frac{1}{2}} d u

\int u^{\frac{5}{2}} d u = \frac{2}{7} u^{\frac{7}{2}}

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

= \frac{2}{7} u^{\frac{7}{2}} - \frac{2}{3} u^{\frac{3}{2}}

u = sec (x)

代回

= \frac{2}{7} sec^{\frac{7}{2}} (x) - \frac{2}{3} sec^{\frac{3}{2}} (x)

解答补常数

= \frac{2}{7} sec^{\frac{7}{2}} (x) - \frac{2}{3} sec^{\frac{3}{2}} (x) + C

\int sin^{6} (x) \cdot cos^{5} (x) d x

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

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

\int sin^{3} (t) cos^{3} (t) d t

\int \frac{5 x}{( x + 4 ) ( x + 2 )} d x