integral of (tan(x)+1/(tan(x)))^2

解答

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

- 2 cot (2 x) + C

求解步骤

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

使用三角恒等式改写

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

乘开

\frac{( sin ^{2} ( x ) + cos ^{2} ( x ) ) ^{2}}{cos ^{2} ( x ) sin ^{2} ( x )} : \frac{sin ^{2} ( x )}{cos ^{2} ( x )} + 2 + \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

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

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

\int 2 d x = 2 x

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

= tan (x) - x + 2 x - cot (x) - x

化简

tan (x) - x + 2 x - cot (x) - x : - 2 cot (2 x)

= - 2 cot (2 x)

解答补常数

= - 2 cot (2 x) + C

x^{2} y^{^{''}} - 2 y = sin (ln (x))

x \to - 2 lim (2 x)

\int \frac{1}{5 x ^{7}} d x

ma c l a u r in \frac{arctan ( x )}{x + 1}

\frac{\partial}{\partial y} (y - cos (y))