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

解答

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

ln ∣ tan (x) + sec (x) ∣ - sin (x) + C

求解步骤

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

化简

\frac{sin ( x ) sin ( x )}{cos ( x )} : sin (x) tan (x)

= \int sin (x) tan (x) d x

使用三角恒等式改写

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

乘开

\frac{1 - cos ^{2} ( x )}{cos ( x )} : \frac{1}{cos ( x )} - cos (x)

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

\int \frac{1}{cos ( x )} d x = ln ∣ tan (x) + sec (x) ∣

\int cos (x) d x = sin (x)

= ln ∣ tan (x) + sec (x) ∣ - sin (x)

解答补常数

= ln ∣ tan (x) + sec (x) ∣ - sin (x) + C

\int - \frac{1}{9 x 81 x ^{2} - 9} d x

\int \frac{1}{x ( 9 + ln ^{2} ( x ) )} d x

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

\int (sec^{2} (x) - 6) d x

\int (4 - x) 3 x d x