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

解答

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

\frac{5}{2} x - \frac{1}{4} sin (2 x) - cot (x) + C

求解步骤

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

使用三角恒等式改写

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

乘开

\frac{( sin ^{2} ( x ) + 1 ) ^{2}}{sin ^{2} ( x )} : sin^{2} (x) + 2 + \frac{1}{sin ^{2} ( x )}

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

\int sin^{2} (x) d x = \frac{1}{2} (x - \frac{1}{2} sin (2 x))

\int 2 d x = 2 x

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

= \frac{1}{2} (x - \frac{1}{2} sin (2 x)) + 2 x - cot (x)

化简

\frac{1}{2} (x - \frac{1}{2} sin (2 x)) + 2 x - cot (x) : \frac{5}{2} x - \frac{1}{4} sin (2 x) - cot (x)

= \frac{5}{2} x - \frac{1}{4} sin (2 x) - cot (x)

解答补常数

= \frac{5}{2} x - \frac{1}{4} sin (2 x) - cot (x) + C

\int \frac{x}{3} e^{- \frac{x}{3}} d x

\int \frac{3 x ^{2} - 2 x + 19}{( x - 1 ) ( x ^{2} + 9 )} d x

\int sin^{2} (8 n - 3) d n

\int (5 e^{x} + \frac{10}{x} - 4 x^{3} + 2^{x}) d x

\int sin^{2} (x) \cdot cos^{7} (x) d x