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

解答

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

\frac{1}{2} ln ∣ tan (x) ∣ + \frac{1}{2} tan (x) + C

求解步骤

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

乘开

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

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

\int \frac{1}{sin ( 2 x )} d x = \frac{1}{2} ln ∣ tan (x) ∣

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

= \frac{1}{2} ln ∣ tan (x) ∣ + \frac{1}{2} tan (x)

解答补常数

= \frac{1}{2} ln ∣ tan (x) ∣ + \frac{1}{2} tan (x) + C

\int 12 x e^{x} d x

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