integral of tan^4(x)*sec^4(x)

解答

\int tan^{4} (x) \cdot sec^{4} (x) d x

\frac{tan ^{5} ( x )}{5} + \frac{tan ^{7} ( x )}{7} + C

求解步骤

\int tan^{4} (x) sec^{4} (x) d x

使用三角恒等式改写

= \int (1 + tan^{2} (x)) sec^{2} (x) tan^{4} (x) d x

使用换元积分法

= \int u^{4} (1 + u^{2}) d u

乘开

u^{4} (1 + u^{2}) : u^{4} + u^{6}

= \int u^{4} + u^{6} 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^{4} d u + \int u^{6} d u

\int u^{4} d u = \frac{u ^{5}}{5}

\int u^{6} d u = \frac{u ^{7}}{7}

= \frac{u ^{5}}{5} + \frac{u ^{7}}{7}

u = tan (x)

代回

= \frac{tan ^{5} ( x )}{5} + \frac{tan ^{7} ( x )}{7}

解答补常数

= \frac{tan ^{5} ( x )}{5} + \frac{tan ^{7} ( x )}{7} + C

\int 18 θ sin (18 θ^{2}) d θ

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

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

\int 10 x^{- 3} d x

\int 2000 (1 + 0.2 x)^{- \frac{3}{2}} d x