integral of tan^7(x)

解答

\int tan^{7} (x) d x

- ln ∣ sec (x) ∣ + \frac{3 sec ^{2} ( x )}{2} - \frac{3 sec ^{4} ( x )}{4} + \frac{sec ^{6} ( x )}{6} + C

求解步骤

\int tan^{7} (x) d x

使用三角恒等式改写

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

使用换元积分法

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

乘开

\frac{( - 1 + u ^{2} ) ^{3}}{u} : - \frac{1}{u} + 3 u - 3 u^{3} + u^{5}

= \int - \frac{1}{u} + 3 u - 3 u^{3} + u^{5} 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 \frac{1}{u} d u + \int 3 u d u - \int 3 u^{3} d u + \int u^{5} d u

\int \frac{1}{u} d u = ln ∣ u ∣

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

\int 3 u^{3} d u = \frac{3 u ^{4}}{4}

\int u^{5} d u = \frac{u ^{6}}{6}

= - ln ∣ u ∣ + \frac{3 u ^{2}}{2} - \frac{3 u ^{4}}{4} + \frac{u ^{6}}{6}

u = sec (x)

代回

= - ln ∣ sec (x) ∣ + \frac{3 sec ^{2} ( x )}{2} - \frac{3 sec ^{4} ( x )}{4} + \frac{sec ^{6} ( x )}{6}

解答补常数

= - ln ∣ sec (x) ∣ + \frac{3 sec ^{2} ( x )}{2} - \frac{3 sec ^{4} ( x )}{4} + \frac{sec ^{6} ( x )}{6} + C

d er i v a t i v e y = (x^{5} + 1)^{6}

d er i v a t i v e y = 4 x^{2} + 2

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

d er i v a t i v e 1 2^{9 x}

x \to 2 lim (\frac{x - 2}{∣ x - 2 ∣})