integral of (sec^2(y))/(csc(y))

解答

\int \frac{sec ^{2} ( y )}{csc ( y )} d y

sec (y) + C

求解步骤

\int \frac{sec ^{2} ( y )}{csc ( y )} d y

使用三角恒等式改写

= \int \frac{1 + tan ^{2} ( y )}{csc ( y )} d y

乘开

\frac{1 + tan ^{2} ( y )}{csc ( y )} : \frac{1}{csc ( y )} + \frac{tan ^{2} ( y )}{csc ( y )}

= \int \frac{1}{csc ( y )} + \frac{tan ^{2} ( y )}{csc ( y )} d y

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}{csc ( y )} d y + \int \frac{tan ^{2} ( y )}{csc ( y )} d y

\int \frac{1}{csc ( y )} d y = - cos (y)

\int \frac{tan ^{2} ( y )}{csc ( y )} d y = cos (y) + sec (y)

= - cos (y) + cos (y) + sec (y)

化简

= sec (y)

解答补常数

= sec (y) + C

\frac{d}{d x} (\frac{3 x}{x ^{2} + 9})

x \to 0 lim (x^{2} - 2 x^{3})

n = 0 \sum \infty \frac{1}{n !} x^{n}

x \to 0 lim (- x^{2} + 4 x)

y^{^{''}} - \frac{3}{x} y^{^{'}} + \frac{4}{x ^{2}} y = 0