integral of (-csc^2(x))/(cot^7(x))

解答

\int \frac{- csc ^{2} ( x )}{cot ^{7} ( x )} d x

- \frac{3}{2} sec^{2} (x) + \frac{3}{4} sec^{4} (x) - \frac{1}{6} sec^{6} (x) + sec^{2} (x) - \frac{1}{4} sec^{4} (x) + C

求解步骤

\int \frac{- csc ^{2} ( x )}{cot ^{7} ( x )} d x

提出常数:

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

= - \int \frac{csc ^{2} ( x )}{cot ^{7} ( x )} d x

使用三角恒等式改写

= - \int \frac{1 + cot ^{2} ( x )}{cot ^{7} ( x )} d x

乘开

\frac{1 + cot ^{2} ( x )}{cot ^{7} ( x )} : \frac{1}{cot ^{7} ( x )} + \frac{1}{cot ^{5} ( x )}

= - \int \frac{1}{cot ^{7} ( x )} + \frac{1}{cot ^{5} ( x )} 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{1}{cot ^{7} ( x )} d x + \int \frac{1}{cot ^{5} ( x )} d x)

\int \frac{1}{cot ^{7} ( x )} d x = - ln ∣ sec (x) ∣ + \frac{3 sec ^{2} ( x )}{2} - \frac{3 sec ^{4} ( x )}{4} + \frac{sec ^{6} ( x )}{6}

\int \frac{1}{cot ^{5} ( x )} d x = ln ∣ sec (x) ∣ - sec^{2} (x) + \frac{sec ^{4} ( x )}{4}

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

化简

- (- ln ∣ sec (x) ∣ + \frac{3 sec ^{2} ( x )}{2} - \frac{3 sec ^{4} ( x )}{4} + \frac{sec ^{6} ( x )}{6} + ln ∣ sec (x) ∣ - sec^{2} (x) + \frac{sec ^{4} ( x )}{4}) : - \frac{3}{2} sec^{2} (x) + \frac{3}{4} sec^{4} (x) - \frac{1}{6} sec^{6} (x) + sec^{2} (x) - \frac{1}{4} sec^{4} (x)

= - \frac{3}{2} sec^{2} (x) + \frac{3}{4} sec^{4} (x) - \frac{1}{6} sec^{6} (x) + sec^{2} (x) - \frac{1}{4} sec^{4} (x)

解答补常数

= - \frac{3}{2} sec^{2} (x) + \frac{3}{4} sec^{4} (x) - \frac{1}{6} sec^{6} (x) + sec^{2} (x) - \frac{1}{4} sec^{4} (x) + C