integral of ((sin(x))/(cos(x)))^2

解答

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

- x + tan (x) + C

求解步骤

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

使用三角恒等式改写

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

\int 1 d x = x

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

= - x + tan (x)

解答补常数

= - x + tan (x) + C

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

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

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

\int 6 x - 3 x^{2} d x

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